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Graph Angles in a Standard Position

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 [more…]

Identify Coterminal Angles

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 [more…]

How to Rename Coterminal Angles

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 [more…]

How to Change Degrees to Radians

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 [more…]

How to Change Radians to Degrees

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 [more…]

How to Relate Radians to Radar Scans

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 [more…]

How to Measure the Distance to the Moon Using 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 [more…]

How to Measure the Speed of a Car around a Race Track

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 [more…]

Define a Right Triangle and Its Parts

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 [more…]

Identify Common Pythagorean Triples

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 [more…]

Use One Function to Solve for Another Function

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? [more…]

How to Remember the Ratios for the Three Basic Trig Functions

The legend of SohCahToa is an easy way to remember the ratios for the three basic trig functions. Sure, the story is pretty lame, but you'll find it very useful when trying to remember the ratios for the [more…]

How to Compare Slice Sizes on Two Pizzas Using 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 [more…]

How to Find the Distance across a Pond

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). [more…]

Using the 30-60-90 Right Triangle

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 [more…]

Using Isosceles Right Triangles

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 [more…]

Labeling the Sides of a Right Triangle

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 [more…]

A Quick Table for the Three Reciprocal Trigonometry Functions

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 [more…]

How to Place Points on a Unit Circle

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 [more…]

How to Compute Reference Angles in Degrees

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 [more…]

How to Compute the Reference Angles in Radians

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 [more…]

How to Calculate Trigonometry Functions of Angles Using the Unit Circle

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. [more…]

How to Calculate Coordinates at the Origin on Any Unit Circle

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 [more…]

Domain and Range of Sine and Cosine Functions

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 [more…]

Angle of Elevation and Angle of Depression in Trigonometry Functions

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 [more…]

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