Trigonometry Articles

Cheat Sheet / Updated 02-09-2023

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.

Article / Updated 08-11-2022

Even though each trigonometry function is perfectly wonderful, being able to express each trig function in terms of one of the other five trig functions is frequently to your advantage. For example, you may have some sine terms in an expression that you want to express in terms of tangent, so that all the functions match, making it easier to solve the equation. To rewrite the sine function in terms of tangent, follow these steps: Start with the ratio identity involving sine, cosine, and tangent, and multiply each side by cosine to get the sine alone on the left. Replace cosine with its reciprocal function. Solve the Pythagorean identity tan2θ + 1 = sec2θ for secant. Replace the secant in the sine equation.

Article / Updated 08-10-2022

Even though each trigonometry function is perfectly wonderful, being able to express each trig function in terms of one of the other five trig functions is frequently to your advantage. For example, you may have some sine terms in an expression that you want to express in terms of cotangent, so that all the functions match, making it easier to solve the equation. To write the sine function in terms of cotangent, follow these steps: Start with the ratio identity involving sine, cosine, and tangent, and multiply each side by cosine to get the sine alone on the left. Replace cosine with its reciprocal function. Solve the Pythagorean identity tan2θ + 1 = sec2θ for secant. Replace the secant in the sine equation. Replace all the tangents with 1 over the reciprocal for tangent (which is cotangent) and simplify the expression. The result is a complex fraction — it has fractions in both the numerator and denominator — so it’ll look a lot better if you simplify it. Rewrite the part under the radical as a single fraction and simplify it by taking the square root of each part. Multiply the numerator by the reciprocal of the denominator. Voilà — you have sine in terms of cotangent.

Article / Updated 12-21-2021

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 is Look at the graphs of the sine and cosine functions on the same coordinate axes, as shown in the following figure. The graph of the cosine is the darker curve; note how it’s shifted to the left of the sine curve. The graphs of the sine and cosine functions illustrate a property that exists for several pairings of the different trig functions. The property represented here is based on the right triangle and the two acute or complementary angles in a right triangle. The identities that arise from the triangle are called the cofunction identities. The cofunction identities are as follows: These identities show how the function values of the complementary angles in a right triangle are related. For example, cosθ = sin (90° – θ) means that if θ is equal to 25 degrees, then cos 25° = sin (90° – 25°) = sin 65°. This equation is a roundabout way of explaining why the graphs of sine and cosine are different by just a slide. You probably noticed that these cofunction identities all use the difference of angles, but the slide of the sine function to the left was a sum. The shifted sine graph and the cosine graph are really equivalent — they become graphs of the same set of points. Here’s how to prove this statement. You want to show that the sine function, slid 90 degrees to the left, is equal to the cosine function: Replace cos x with its cofunction identity. Apply the two identities for the sine of the sum and difference of two angles. The two identities are Substituting in the x’s and angles, Simplify the terms by using the values of the functions. So you see, the shifted sine graph is equal to the cosine graph.

Article / Updated 07-12-2021

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. The figure shows a circle with a radius of r that has an angle drawn in standard position. The equation of a circle is x2 + y2 = r2. Based on this equation and the coordinates of the point, (x,y), where the terminal side of the angle intersects the circle, the six trig functions for angle theta are defined as follows: You can see where these definitions come from if you picture a right triangle formed by dropping a perpendicular segment from the point (x,y) to the x-axis. The following figure shows such a right triangle. Remember that the x-value is to the right (or left) of the origin, and the y-value is above (or below) the x-axis — and use those values as lengths of the triangle’s sides. Therefore, the side opposite angle theta is y, the value of the y-coordinate. The adjacent side is x, the value of the x-coordinate. Take note that for angles in the second quadrant, for example, the x-values are negative, and the y-values are positive. The radius, however, is always a positive number. With the x-values negative and the y-values positive, you see that the sine and cosecant are positive, but the other functions are all negative, because they all have an x in their ratios. The signs of the trig functions all fall into line when you use this coordinate system, so no need to worry about remembering the ASTC rule here.

Article / Updated 07-09-2021

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. The measure of the central angle is the same as the measure of the arc that the two sides cut out of the circle. Inscribed angle An inscribed angle has its vertex on the circle, and the sides of the angle lie on two chords of the circle. The measure of the inscribed angle is half that of the arc that the two sides cut out of the circle. Interior angle An interior angle has its vertex at the intersection of two lines that intersect inside a circle. The sides of the angle lie on the intersecting lines. The measure of an interior angle is the average of the measures of the two arcs that are cut out of the circle by those intersecting lines. Exterior angle An exterior angle has its vertex where two rays share an endpoint outside a circle. The sides of the angle are those two rays. The measure of an exterior angle is found by dividing the difference between the measures of the intercepted arcs by two. Example: Find the measure of angle EXT, given that the exterior angle cuts off arcs of 20 degrees and 108 degrees. Find the difference between the measures of the two intercepted arcs and divide by 2: The measure of angle EXT is 44 degrees. Sectioning sectors A sector of a circle is a section of the circle between two radii (plural for radius). You can consider this part like a piece of pie cut from a circular pie plate. You can find the area of a sector of a circle if you know the angle between the two radii. A circle has a total of 360 degrees all the way around the center, so if that central angle determining a sector has an angle measure of 60 degrees, then the sector takes up 60/360 or 1/6, of the degrees all the way around. In that case, the sector has 1/6 the area of the whole circle. Example: Find the area of a sector of a circle if the angle between the two radii forming the sector is 80 degrees and the diameter of the circle is 9 inches. Find the area of the circle. The area of the whole circle is or about 63.6 square inches. Find the portion of the circle that the sector represents. The sector takes up only 80 degrees of the circle. Divide 80 by 360 to get Calculate the area of the sector. Multiply the fraction or decimal from Step 2 by the total area to get the area of the sector: The whole circle has an area of almost 64 square inches, and the sector has an area of just over 14 square inches.

Article / Updated 07-09-2021

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. Use the Pythagorean theorem to solve for the missing length. Replace the variables in the theorem with the values of the known sides. 482 + 142 = c2 Square the measures and add them together. The length of the missing side, c, which is the hypotenuse, is 50. The triangle on the right is missing the bottom length, but you do have the length of the hypotenuse. It doesn't matter whether you call the missing length a or b. Replace the variables in the theorem with the values of the known sides. 332 + b2 = 1832 Square the measures, and subtract 1,089 from each side. Find the square root of each side. The length of the missing side is 180 units. That's not much shorter than the hypotenuse, but it still shows that the hypotenuse has the longest measure.

Article / Updated 07-07-2021

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. Positive angles The positive angles on the unit circle are measured with the initial side on the positive x-axis and the terminal side moving counterclockwise around the origin. The figure shows some positive angles labeled in both degrees and radians. Notice that the terminal sides of the angles measuring 30 degrees and 210 degrees, 60 degrees and 240 degrees, and so on form straight lines. This fact is to be expected because the angles are 180 degrees apart, and a straight angle measures 180 degrees. You see the significance of this fact when you deal with the trig functions for these angles. Negative angles Just when you thought that angles measuring up to 360 degrees or 2π radians was enough for anyone, you’re confronted with the reality that many of the basic angles have negative values and even multiples of themselves. If you measure angles clockwise instead of counterclockwise, then the angles have negative measures: A 30-degree angle is the same as an angle measuring –330 degrees, because they have the same terminal side. Likewise, an angle of is the same as an angle of But wait — you have even more ways to name an angle. By doing a complete rotation of two (or more) and adding or subtracting 360 degrees or a multiple of it before settling on the angle’s terminal side, you can get an infinite number of angle measures, both positive and negative, for the same basic angle. For example, an angle of 60 degrees has the same terminal side as that of a 420-degree angle and a –300-degree angle. The figure shows many names for the same 60-degree angle in both degrees and radians. Although this name-calling of angles may seem pointless at first, there’s more to it than arbitrarily using negatives or multiples of angles just to be difficult. The angles that are related to one another have trig functions that are also related, if not the same.

Article / Updated 07-07-2021

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. With the measure of the radius, you can tell a lot about the circle: its diameter (the distance from one side to the other, passing through the center), its circumference (how far around it is), and its area (how many square inches, feet, yards, meters — what have you — fit into it). Ancient mathematicians figured out that the circumference of a circle is always a little more than three times the diameter of a circle. Since then, they narrowed that “little more than three times” to a value called pi (pronounced “pie”), designated by the Greek letter π. The decimal value of π isn't exact — it goes on forever and ever, but most of the time, people refer to it as being approximately 3.14 or 22/7, whichever form works best in specific computations. The formula for figuring out the circumference of a circle is tied to π and the diameter: Circumference of a circle: C = πd = 2πr The d represents the measure of the diameter, and r represents the measure of the radius. The diameter is always twice the radius, so either form of the equation works. Similarly, the formula for the area of a circle is tied to π and the radius: Area of a circle: A = πr2 This formula reads, “Area equals pi are squared.” Find the radius, circumference, and area of a circle if its diameter is equal to 10 feet in length. If the diameter (d) is equal to 10, you write this value as d = 10. The radius is half the diameter, so the radius is 5 feet, or r = 5. You can find the circumference by using the formula So, the circumference is about 31.5 feet around. You find the area by using the formula so the area is about 78.5 square feet.

Step by Step / Updated 03-27-2016

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.