Trigonometry For Dummies
Overview
Make trigonometry as easy as 1-2-3
Believe it or not, trigonometry is easier than it looks! With the right help, you can breeze through your next trig class, test, or exam and be ready for your next math challenge. In Trigonometry For Dummies, you’ll learn to understand the basics of sines, cosines, and tangents, graph functions, solve tough formulas, and even discover how to use trig outside the classroom in some cool and interesting ways.
Ditch the confusing jargon and take a plain-English tour of one of the most useful disciplines in math. In this lifesaving guide, you’ll learn how to:
- Graph trig functions, including sine, cosine, tangent, and cotangent functions
- Understand inverse trig functions and solve trig equations
- Relate triangles to circular functions and get a handle on basic identities
So, whether you’re looking for an easy-to-use study guide, to boost your math grade, or get a refresher on some basic trig concepts after a long absence from studying, Trigonometry For Dummies is your ticket to understanding the mathematical mysteries of the triangle.
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About The Author
Mary Jane Sterling is the author of Algebra I For Dummies, Pre-Calculus Workbook For Dummies, Algebra II For Dummies, and oodles of other Dummies titles. She was a Professor of Mathematics at Bradley University in Peoria, Illinois, for more than 35 years, teaching algebra, business calculus, geometry, and finite mathematics.
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trigonometry for dummies
CHEAT SHEET
Trigonometry is the study of triangles, which contain angles, of course. Get to know some special rules for angles and various other important functions, definitions, and translations. Sines and cosines are two trig functions that factor heavily into any study of trigonometry; they have their own formulas and rules that you’ll want to understand if you plan to study trig for very long.
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Trigonometry
A trigonometric function has input values that are angles and output values that are real numbers. You have two choices when inputting values into a trig function: You can use degrees or radians. Most people are more comfortable with the degree measures, but the radian measures have a huge benefit: They’re real numbers — they have decimal values and are multiples of π.
Trigonometry
You know that the reciprocal functions have values that are reciprocals, or flips, of the values for their respective functions. The reciprocal of sine is cosecant, so each function value for cosecant is the reciprocal of sine’s. The same goes for the other two reciprocal functions.
The table shows the reciprocal in each case, in their simplified forms.
Trigonometry
Mathematical problems that require the use of trig functions often have one of two related angles: the angle of elevation or the angle of depression. The scenarios that use these angles usually involve calculating distances that can't be physically measured.
For example, these angles are used when finding the distance from an airplane to a point on the ground or the distance up to a balloon or another object above you.
There are several ways of drawing an angle in a circle, and each has a special way of computing the size of that angle. Four different types of angles are: central, inscribed, interior, and exterior. Here, you see examples of these different types of angles.
Central angle
A central angle has its vertex at the center of the circle, and the sides of the angle lie on two radii of the circle.
Trigonometry
The sine values for 30, 150, 210, and 330 degrees are 1/2, 1/2, –1/2, and –1/2, respectively. All these multiples of 30 degrees have a sin whose absolute value of 1/2 . The following rule and figure help you determine whether a trig-function value is positive or negative. First, note that each quadrant in the figure is labeled with a letter.
When two lines, segments, or rays touch or cross one another, they form an angle or angles. In the case of two intersecting lines, the result is four different angles. When two segments intersect, they can form one, two, or four angles; the same goes for two rays.
These examples are just some of the ways that you can form angles.
Segments, rays, and lines are some of the basic forms found in geometry, and they're almost as important in trigonometry. You use those segments, rays, and lines to form angles.
Drawing segments, rays, and lines
A segment is a straight figure drawn between two endpoints. You usually name it by its endpoints, which you indicate by capital letters.
All on their own, angles are certainly very exciting. But put them into a triangle, and you've got icing on the cake. Triangles are one of the most frequently studied geometric figures. The angles that make up the triangle give them many of their characteristics.
Angles in triangles
A triangle always has three angles.
You show the diameter and radius of a circle by drawing segments from a point on the circle either to or through the center of the circle. But two other straight figures have a place on a circle. One of these figures is called a chord, and the other is a tangent:
Chords: A chord of a circle is a segment that you draw from one point on the circle to another point on the circle.
Trigonometry
The relationship between the cosine and sine graphs is that the cosine is the same as the sine — only it’s shifted to the left by 90 degrees, or π/2. The trigonometry equation that represents this relationship isLook at the graphs of the sine and cosine functions on the same coordinate axes, as shown in the following figure.
Trigonometry
The trig functions can be defined using the measures of the sides of a right triangle. But they also have very useful definitions using the coordinates of points on a graph. First, let let the vertex of an angle be at the origin — the point (0,0) — and let the initial side of that angle lie along the positive x-axis and the terminal side be a rotation in a counterclockwise motion.
A right triangle has a right angle in it. But it can only have one right angle, because the total number of degrees in a triangle is 180. If it had two right angles, then those two angles would take up all 180 degrees; no degrees would be left for a third angle. So in a right triangle, the other two angles share the remaining 90 degrees.
Every triangle has six parts: three sides and three angles. If you measure the sides and then pair up those measurements (taking two at a time), you have three different pairings. Do division problems with the pairings — changing the order in each pair — and you have six different answers. These six different answers represent the six trig functions.
Trigonometry
As you study trigonometry, you'll find occasions when you need to change degrees to radians, or vice versa. A formula for changing from degrees to radians or radians to degrees is:
The formula works for any angle, but the most commonly used angles and their equivalences are shown below.
Trigonometry
A function consists of a rule that you apply to the input values. The result is a single output value. You can usually use a huge number of input values, and they're all part of the domain of the function. The output values make up the range of the function.
Domain of a trig function
The domain of a function consists of all the values that you can use as input into the function rule.
Trigonometry
Here's an application of trigonometry that you may very well be able to relate to: Have you ever tried to get a large piece of furniture around a corner in a house? You twist and turn and put it up on end, but to no avail. In this example, pretend that you're trying to get a 15-foot ladder around a corner where two 4-foot-wide hallways meet at a 90-degree angle.
Trigonometry
The cosecant and secant functions are closely tied to sine and cosine, because they're the respective reciprocals. In reference to the coordinate plane, cosecant is r/y, and secant is r/x. The value of r is the length of the hypotenuse of a right triangle — which is always positive and always greater than x and y.
Trigonometry
The sine and cosine functions are unique in the world of trig functions, because their ratios always have a value. No matter what angle you input, you get a resulting output. The value you get may be 0, but that’s a number, too. In reference to the coordinate plane, sine is y/r, and cosine is x/r.
The radius, r, is always some positive number (which is why these functions always have a value, because they don’t ask you to divide by 0), and r is always a number greater than (or equal to) the absolute value of x or y.
Trigonometry
The tangent and cotangent are related not only by the fact that they’re reciprocals, but also by the behavior of their ranges. In reference to the coordinate plane, tangent is y/x, and cotangent is x/y. The domains of both functions are restricted, because sometimes their ratios could have zeros in the denominator, but their ranges are infinite.
Trigonometry
The domain of a function consists of all the input values that a function can handle — the way the function is defined. Of course, you want to get output values (which make up the range) when you enter input values.
But sometimes, when you input something that doesn’t belong in the function, you end up with some impossible situations.
Trigonometry
Using the lengths of the sides of the two special right triangles — the 30-60-90 right triangle and the 45-45-90 right triangle — the following exact values for trig functions are found. Using these values in conjunction with reference angles and signs of the functions in the different quadrants, you can determine the exact values of the multiples of these angles.
Trigonometry
You can determine the trig functions for any angles found on the unit circle — any that are graphed in standard position (meaning the vertex of the angle is at the origin, and the initial side lies along the positive x-axis). You use the rules for reference angles, the values of the functions of certain acute angles, and the rule for the signs of the functions.
You probably know many of the trigonometry functions for the more common angles. Some favorites are:
cos 0° = 1, and tan 45° = 1. That’s three values for three different angles in three different functions. This doesn’t even scratch the surface. Trig functions produce numerical values for all angles, whether in degrees or radians.
Many of the formulas used in trigonometry are also found in algebra and analytic geometry. But trigonometry also has some special formulas usually found just in those discussions. A formula provides you a rule or equation that you can count on to work, every single time. A formula gives a relationship between particular quantities and units.
Navigators, surveyors, and carpenters all use the same angle measures, but the angles start out in different positions or places. In trigonometry and most other mathematical disciplines, you draw angles in a standard, universal position, so that mathematicians around the world are drawing and talking about the same thing.
Trigonometry
You don’t need a unit circle to use this coordinate business when determining the function values of angles graphed in standard position on a circle. You can use a circle with any radius, as long as the center is at the origin. The standard equation for a circle centered at the origin is x2 + y2 = r2.
Using the angles shown, find the sine of alpha.
Trigonometry
Calculating trig functions of angles within a unit circle is easy as pie. The figure shows a unit circle, which has the equation x2 + y2 = 1, along with some points on the circle and their coordinates.
Using the angles shown, find the tangent of theta.
Find the x- and y-coordinates of the point where the angle’s terminal side intersects with the circle.
The computation required for changing degrees to radians isn't difficult. The computation involves a few tricks, though, and the format is important. You don't usually write the radian measures with decimal values unless you've multiplied through by the decimal equivalent for pi.
To change a measure in degrees to radians, start with the basic proportion for the equivalent angle measures:
For example, here's how you change a measure of 40 degrees to radians:
Put the 40 in place of the first numerator in the proportion.
You use the same basic proportion to change radians to degrees as you do for changing degrees to radians. The computation required for changing radians to degrees isn't difficult. For example, to change
radians to a degree measure:
Put the radian measure in the proportion.
Simplify the complex fraction on the right by multiplying the numerator by the reciprocal of the denominator.
Every triangle can be circumscribed by a circle, meaning that one circle — and only one — goes through all three vertices (corners) of any triangle. In laymen's terms, any triangle can fit into some circle with all its corners touching the circle.
To circumscribe a triangle, all you need to do is find the circumcenter of the circle (at the intersection of the perpendicular bisectors of the triangle's sides).
Trigonometry
Some fraternity brothers want to order pizza — and you know how hungry college men can be. The big question is, which has bigger slices of pizza: a 12-inch pizza cut into six slices, or a 15-inch pizza cut into eight slices?
The figure shows a 12-inch pizza and a 15-inch pizza, both of which are sliced. Can you tell by looking at them which slices are bigger — that is, have more area?
Solving for the reference angle in degrees is much easier than trying to determine a trig function for the original angle. To compute the measure (in degrees) of the reference angle for any given angle theta, use the rules in the following table.
Finding Reference Angles in Degrees
Quadrant
Measure of Angle Theta
Measure of Reference Angle
I
0° to 90°
theta
II
90° to 180°
180° – theta
III
180° to 270°
theta – 180°
IV
270° to 360°
360° – theta
Find the reference angle for 200 degrees:
Determine the quadrant in which the terminal side lies.
Trigonometry
Solving for the reference angle in radians is much easier than trying to determine a trig function for the original angle. To compute the measure (in radians) of the reference angle for any given angle theta, use the rules in the following table.
Finding Reference Angles in Radians
Quadrant
Measure of Angle Theta
Measure of Reference Angle
I
0 to π/2
è
II
π/2 to π
ð – è
III
π to 3π/2
è – ð
IV
3π/2 to 2π
2ð – è
To find the reference angle for
Determine the quadrant in which the terminal side lies.
Trigonometry
The angles used most often in trig have trig functions with convenient exact values. Other angles don’t cooperate anywhere near as nicely as these popular ones do. A quick, easy way to memorize the exact trig-function values of the most common angles is to construct a table, starting with the sine function and working with a pattern of fractions and radicals.
You can use trigonometry functions to determine the altitude of a balloon. Cindy and Mindy, standing a mile apart, spot a hot-air balloon directly above a particular point on the ground somewhere between them. The angle of elevation from Cindy to the balloon is 60 degrees; the angle of elevation from Mindy to the balloon is 70 degrees.
Which trig function should you use to determine the height of a tree? Suppose you're flying a kite, and it gets caught at the top of a tree. You've let out all 100 feet of string for the kite, and the angle that the string makes with the ground (the angle of elevation) is 75 degrees.
Instead of worrying about how to get your kite back, you wonder, “How tall is that tree?
Trigonometry
Trig functions come in handy if you work for NASA or need to measure the vertical distance travelled by a rocket. In this example, a rocket is shot off and travels vertically as a scientist, who's a mile away, watches its flight. One second into the flight, the angle of elevation of the rocket is 30 degrees. Two seconds later, the angle of elevation is 60 degrees.
Trigonometry
If you can find the midpoint of a segment, you can divide it into two equal parts. Finding the middle of each of the two equal parts allows you to find the points needed to divide the entire segment into four equal parts. Finding the middle of each of these segments gives you eight equal parts, and so on.
For example, to divide the segment with endpoints (–15,10) and (9,2) into eight equal parts, find the various midpoints like so:
The midpoint of the main segment from (–15,10) to (9,2) is (–3,6).
Trigonometry
If you have the value of one of a point’s coordinates on the unit circle and need to find the other, you can substitute the known value into the unit-circle equation and solve for the missing value.
You can choose any number between 1 and –1, because that’s how far the unit circle extends along the x- and y-axes.
The trig functions all have inverses, but only under special conditions — you have to restrict the domain values. Not all functions have inverses, and not all inverses are easy to determine. Here's a nice method for finding inverses of basic algebraic functions.
Use algebra to find an inverse function
The most efficient method for finding an inverse function for a given one-to-one function involves the following steps:
Replace the function notation name with y.
Trigonometry is very handy for finding distances that you can’t reach to measure. Imagine that you want to string a cable diagonally across a pond (so you can attach a bunch of fishing line and hooks). The diagonal distance is the hypotenuse of a right triangle. You can measure the other two sides along the shore.
The middle of a line segment is its midpoint. To find the midpoint of a line segment, you just calculate the averages of the coordinates — easy as pie.
The midpoint, M, of a segment with endpoints (x1,y1) and (x2,y2) is
If you want to know the midpoint of the segment with endpoints (–4,–1) and (2,5), then plug the numbers into the midpoint formula, and you get a midpoint of (–1,2):
See how this segment looks in graph form in the following figure.
Each of the angles in a unit circle has a reference angle, which is always a positive acute angle (except the angles that are already positive and acute). By identifying the reference angle, you can determine the function values for that reference angle and, ultimately, the original angle.
Usually, solving for the reference angle first is much easier than trying to determine a trig function for the original angle.
One way to describe the middle of a circle is to identify the centroid. This middle-point is the center of gravity, where you could balance the triangle and spin it around.
When you graph a circle, triangle, or line segment by using coordinate axes, then you can name these middle points with a pair of x- and y-coordinates.
Trigonometry
You can use trig functions to measure the distance between the rooftops on buildings. Why would you need to do this? Well, Jumping Jehoshaphat makes his living by jumping, on his motorcycle, from building to building, cliff to bluff, or any place he can get attention for doing it. His record jump is a distance of 260 feet, from one building to another.
Trigonometry
One of the earliest applications of trigonometry was in measuring distances that you couldn't reach, such as distances to planets or the moon or to places on the other side of the world. Consider the following example.
The diameter of the moon is about 2,160 miles. When the moon is full, a person sighting the moon from the earth measures an angle of 0.
Land surveyors use trigonometry and their fancy equipment to measure things like the slope of a piece of land (how far it drops over a certain distance). Have you ever noticed a worker along the road, peering through an instrument, looking at a fellow worker holding up a sign or a flag?
Haven't you ever wondered what they're doing?
Trigonometry
A race car is going around a circular track. A photographer standing at the center of the track takes a picture, turns 80 degrees, and then takes another picture 10 seconds later. If the track has a diameter of 1/2 mile, how fast is the race car going? The figure shows the photographer in the middle and the car in the two different positions.
If you draw lines from each corner (or vertex) of a triangle to the midpoint of the opposite sides, then those three lines meet at a center, or centroid, of the triangle. The centroid is the triangle's center of gravity, where the triangle balances evenly. The coordinates of the centroid are also two-thirds of the way from each vertex along that segment.
The unit circle is a circle with its center at the origin of the coordinate plane and with a radius of 1 unit. Any circle with its center at the origin has the equation x2 + y2 = r2, where r is the radius of the circle. In the case of a unit circle, the equation is x2 + y2 = 1.
This equation shows that the points lying on the unit circle have to have coordinates (x- and y-values) that, when you square each of them and then add those values together, equal 1.
The graphs of the trig functions have many similarities and many differences. The graphs of the sine and cosine look very much alike, as do the tangent and cotangent, and then the secant and cosecant have similarities. But those three groupings do look different from one another.
The one characteristic that ties them all together is the fact that they're periodic, meaning they repeat the same curve or pattern over and over again, in either direction along the x-axis.
Trigonometry
Two lines are parallel if they have the same slope. Two lines are perpendicular if their slopes are negative reciprocals of one another. Numbers that are negative reciprocals have a product –1.
Consider the following slopes of some lines or line segments:
Here are the slopes of the lines that are parallel:
m1 = 1/2 and m4 = 5/10 have the same slope.
The biggest advantage of using radians instead of degrees is that a radian is directly tied to a length — the length or distance around a circle, which is called its circumference. Using radians is very helpful when doing applications involving the length of an arc of a circle.
Radar scans a circular area that has a radius of 40 miles.
Any angle can have many, many descriptions in terms of angle measures, because an angle is equivalent to its coterminal angles. The most frequently used positive angle measures are those that measure between 0 and 360 degrees. Rules for coterminal angles involve adding or subtracting rotations (or multiples of 360 degrees).
Trigonometry
The Pythagorean theorem states that a2 + b2 = c2 in a right triangle where c is the longest side. You can use this equation to figure out the length of one side if you have the lengths of the other two. The figure shows two right triangles that are each missing one side's measure.In the left triangle, the measure of the hypotenuse is missing.
Defining a function or explaining how it works can involve a lot of words and can get rather lengthy and awkward. Imagine having to write, "Square the input, multiply that result by 2, and then subtract 3." Mathematicians are an efficient lot, and they prefer a more precise, quicker way of writing their instructions.
Trigonometry
You can find the area of a triangle using Heron's Formula. Heron's Formula is especially helpful when you have access to the measures of the three sides of a triangle but can't draw a perpendicular height or don't have a protractor for measuring an angle.
Consider the situation where you have a large ball of string that's 100 yards long and you're told to mark off a triangular area — with the string as the marker for the border of the area.
Trigonometry
Sometimes, finding a measure isn't so easy. You may have to deal with an irregular shape, like a triangle, or even calculate your way around a fixed object. Whatever the case, you can use trigonometry to find the answers you've been searching for.
The most commonly used formula for the area of a triangle is
where A is the area, b is the length of the triangle's base, and h is the height of the triangle drawn perpendicular to that base.
A Pythagorean triple is a list of three numbers that works in the Pythagorean theorem — the square of the largest number is equal to the sum of the squares of the two smaller numbers. The multiple of any Pythagorean triple (multiply each of the numbers in the triple by the same number) is also a Pythagorean triple.
Two angles are coterminal if they have the same terminal side. You have an infinite number of ways to give an angle measure for a particular terminal ray. Sometimes, using a negative angle rather than a positive angle is more convenient, or the answer to an application may involve more than one revolution (spinning around and around).
Trigonometry
Solving identities is almost a rite of passage for those studying trigonometry. Tackling the prospect of solving identities — and later simplifying trig expressions in calculus — goes much more smoothly if you have some algebraic tools at hand. With a plan of action, you’ll succeed more quickly and efficiently and have the desired product.
A right triangle has two shorter sides, or legs, and the longest side, opposite the right angle, which is always called the hypotenuse. The two shorter sides have some other special names, too, based on which acute angle of the triangle you happen to be working with at a particular time.
In reference to acute angle θ, the leg on the other side of the triangle from θ is called the opposite side.
The laws of sines and cosines give you relationships between the lengths of the sides and the trig functions of the angles. These laws are used when you don’t have a right triangle — they work in any triangle. You determine which law to use based on what information you have. In general, the side a lies opposite angle A, the side b is opposite angle B, and side c is opposite angle C.
Trigonometry
The unit circle is a platform for describing all the possible angle measures from 0 to 360 degrees, all the negatives of those angles, plus all the multiples of the positive and negative angles from negative infinity to positive infinity. In other words, the unit circle shows you all the angles that exist.Because a right triangle can only measure angles of 90 degrees or less, the circle allows for a much-broader range.
The trig product-to-sum identities look very much alike. You have to pay close attention to the subtle differences so that you can apply them correctly. Even though the product looks nice and compact, it’s not always as easy to deal with in calculus computations — the sum or difference of two different angles is preferred.
Trigonometry
A circle is a geometric figure that needs only two parts to identify it and classify it: its center (or middle) and its radius (the distance from the center to any point on the circle). After you've chosen a point to be the center of a circle and know how far that point is from all the points that lie on the circle, you can draw a fairly decent picture.
A big advantage of trig expressions and equations is that you can adjust them in so many ways to suit your needs. The basic reciprocal identities here are the ones people use most frequently.
Take a look at the first reciprocal identity and its counterpart:
and
An alternate way of writing these identities uses an exponent of ‒1 rather than a fraction:
Note that the exponents apply to the entire function.
Trigonometry
Two types of transformations act like reflections or flips. One transformation changes all positive outputs to negative and all negative outputs to positive. The other reverses the inputs — positive to negative and negative to positive.
Reflecting up and down (outputs changed): –f(x)
Reflecting left and right (inputs changed): f(–x)
The figure shows reflections of the function
Reflecting downward puts all the points below the x-axis.
Trigonometry
The basic trig functions can be defined with ratios created by dividing the lengths of the sides of a right triangle in a specific order. The label hypotenuse always remains the same — it’s the longest side. But the designations of opposite and adjacent can change — depending on which angle you’re referring to at the time.
Trigonometry
An angle is in standard position when its vertex is at the origin, its initial side is on the positive x-axis, and the terminal side rotates counterclockwise from the initial side. The position of the terminal side determines the sign of the various trig functions of that angle. The following shows you which functions are positive — and you can assume that the other functions are negative in that quadrant.
You can approximate, fairly accurately, the sine and cosine of angles with an infinite series, which is the sum of the terms of some sequence, or list, of numbers. Take note, however, that the series for sine and cosine are accurate only for angles from about –90 degrees to 90 degrees.
The series for the sine of an angle is
and the series for the cosine of an angle is
To use these formulas, you have to write the angle measure, x, in radians and carry out the computations several places.
Every right triangle has the property that the sum of the squares of the two legs is equal to the square of the hypotenuse (the longest side). The Pythagorean theorem is written: a2 + b2 = c2. What’s so special about the two right triangles shown here is that you have an even more special relationship between the measures of the sides — one that goes beyond (but still works with) the Pythagorean theorem.
The sum to product identities are useful for modeling what happens with sound frequencies. Think of two different tones represented by sine curves. Add them together, and they beat against each other with a warble — how much depends on their individual frequencies. The identities give a function modeling what’s happening.
The cosecant function, abbreviated csc, is the reciprocal of the sine function and thus uses this ratio: hypotenuse/opposite. The hypotenuse of a right triangle is always the longest side, so the numerator of this fraction is always larger than the denominator. As a result, the cosecant function always produces values bigger than 1.
Trigonometry
The trig function cosine, abbreviated cos, works by forming this ratio: adjacent/hypotenuse. In the figure, you see that the cosines of the two angles are as follows:
The situation with the ratios is the same as with the sine function — the values are going to be less than or equal to 1 (the latter only when your triangle is a single segment or when dealing with circles), never greater than 1, because the hypotenuse is the denominator.
The last reciprocal function is the cotangent, abbreviated cot. This function is the reciprocal of the tangent (hence, the co-). The ratio of the sides for the cotangent is adjacent/opposite.
You can see that the two cotangents are
The ratio for the cotangent is just that ratio, not necessarily the lengths of the sides.
The most familiar graphing plane is the one using the Cartesian coordinates. You find two perpendicular lines called axes. The horizontal axis (or x-axis) and vertical axis (or y-axis) intersect at the origin, the point labeled (0, 0). The points on this Cartesian plane are labeled with an ordered pair (x,y) where the two coordinates represent the positive or negative distance from the origin, parallel to their respective axis.
When you're using right triangles to define trig functions, the trig function sine, abbreviated sin, has input values that are angle measures and output values that you obtain from the ratio opposite/hypotenuse. The figure shows two different acute angles, and each has a different value for the function sine.
The two values are
The sine is always the measure of the opposite side divided by the measure of the hypotenuse.
Trigonometry
The third trig function, tangent, is abbreviated tan. This function uses just the measures of the two legs and doesn’t use the hypotenuse at all. The tangent is described with this ratio: opposite/adjacent. No restriction or rule on the respective sizes of these sides exists — the opposite side can be larger, or the adjacent side can be larger.
Trigonometry
A translation is a slide, which means that the function has the same shape graphically, but the graph of the function slides up or down or slides left or right to a different position on the coordinate plane.
Sliding up or down
The figure shows the parabola y = x2 with a translation 5 units up and a translation 7 units down.
Trigonometry is the study of triangles, which contain angles, of course. Get to know some special rules for angles and various other important functions, definitions, and translations. Sines and cosines are two trig functions that factor heavily into any study of trigonometry; they have their own formulas and rules that you’ll want to understand if you plan to study trig for very long.
Each of the three sides of a right triangle — hypotenuse, opposite, and adjacent — has a respective length or measure. And those three lengths or measures form six different ratios. Check out the following figure, which has sides of lengths 3, 4, and 5.
The six different ratios that you can form with the numbers 3, 4, and 5 are
These six fractions are all that you can make by using the three lengths of the sides.
Trigonometry
One way to find the values of the trig functions for angles is to use the coordinates of points on a circle that has its center at the origin. Letting the positive x-axis be the initial side of an angle, you can use the coordinates of the point where the terminal side intersects with the circle to determine the trig functions.
Trigonometry
Sometimes you have to solve for a trig function in terms of another function. In the following example, the cosine of angle lambda is 12/13. What are the values of the sine and tangent of lambda?
Identify the sides given by the cosine function.
The cosine ratio is adjacent/hypotenuse. Using the given ratio, the adjacent side measures 12 units, and the hypotenuse measures 13 units.
Trigonometry
Consider a satellite that orbits earth at an altitude of 750 miles. Earth has a radius of 3,950 miles. How far in any direction can the satellite's cameras see? The figure shows the satellite and the length of the camera's scope due to the curvature of earth.
Identify the parts of the triangle that you can use to solve the problem.
The isosceles right triangle, or the 45-45-90 right triangle, is a special right triangle. The two acute angles are equal, making the two legs opposite them equal, too. What’s more, the lengths of those two legs have a special relationship with the hypotenuse (in addition to the one in the Pythagorean theorem, of course).
Trigonometry
Every day, people use trigonometry to measure things that they can't reach. How high is that building? Will this ladder reach to the top of that tree? By using the appropriate trig functions, you can find answers to such questions.
Consider the oh-so-common scenario: A damsel is in distress and is being held captive in a tower.
A 30-60-90 right triangle has angles measuring just what the name says. The two acute, complementary angles are 30 and 60 degrees. These triangles are great to work with, because the angle measures, all being multiples of 30, have a pattern, and so do the measures of the sides.
Oh, yes, the Pythagorean theorem still holds — you have that relationship between the squares of the sides.
What's a degree? In trigonometry, a degree is a tiny slice of a circle. Imagine a pizza cut into 360 equal pieces (what a mess). Each little slice represents one degree.
The first quadrant is the upper right-hand corner of the coordinate plane. That first quadrant is 1/4 of the entire plane. So, if a full circle with its center at the origin has a total of 360 degrees, then 1/4 of it has 90 degrees, which is the measure of the angle that the first quadrant forms.
Trigonometry
Trigonometry is a subject that has to be studied after some background in geometry. Why? Because trigonometry has its whole basis in triangles and angle measures and circles. Geometric studies acquaint you with properties of triangles that are necessary to understand trig concepts — everything makes more sense.
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