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Trigonometric Functions

Using Right Triangle Trigonometry

Trigonometry is the study of the relationships between the angles and sides in triangles. These relationships can be used to calculate different characteristics of triangles.

Concept

Sides in a Right Triangle

In a right triangle, the hypotenuse is always the longest side of the triangle and lies across from the right angle. The triangle's other two sides are called legs. If one of the acute angles in the triangle is labeled, the legs can be discussed relative to that angle. If is labeled as shown, the side that forms the angle is the adjacent side and the side not touching the angle is the opposite side.

Change angles

Note that the opposite and adjacent sides change when changes, but the hypotenuse is always the same.

Concept

Trigonometric Functions

By using the sides of a right triangle, six trigonometric functions of the acute angle can be defined. The most common of these are sine, cosine, and tangent.
Trigonometric Functions

Three other trigonometric functions are cosecant, secant, and cotangent. These are called reciprocal trigonometric functions. It is worth mentioning that these functions must not be confused with the inverse trigonometric functions, which are defined entirely differently.

Reciprocal Trigonometric Functions

The trigonometric functions only state a ratio between two sides in a right triangle; they do not give the lengths of the respective sides. However, if one acute angle and one side are known, the lengths of the other sides can be determined.

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Exercise

Evaluate the trigonometric functions for the angle

Show Solution
Solution

To begin, we should label the triangle's sides. The longest side is the hypotenuse, the side next to is the adjacent side and the side across from is the opposite side.

We can now evaluate the trigonometric functions by using the definitions and the lengths of the sides in the triangle.

trig function definition value

Rule

Pythagorean Theorem

For a right triangle with legs and and hypotenuse the following is true.

This can be proven using area.

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Exercise

For the right triangle, find and

Show Solution
Solution

To begin, we should note that since the given triangle is a right triangle, both the trigonometric ratios and the Pythagorean Theorem might be useful tools. The given information includes one angle and one side. Since two side lengths must be known to use the Pythagorean Theorem, we'll have to use a trigonometric ratio. We can label the sides of the triangle relative to the labeled angle.

Let's first find Since the adjacent side is given, we must use to determine
The length of the hypotenuse is approximately We now know the lengths of two sides in the right triangle. The third side, can be calculated in the same way using or by using the Pythagorean Theorem. We'll use the latter. Remember that is the hypotenuse. The values of and can be assigned arbitrarily.

Thus, the two unknown sides in the right triangle have the lengths

Concept

Inverse Trigonometric Functions

If the value of a trigonometric ratio for a specific angle is known, it is possible to calculate the measure of the angle using inverse trigonometric functions. The inverse function of sine is called inverse sine and is written as

The rest of the inverse trigonometric functions are defined in a similar way.

Function Inverse Function

As long as the appropriate sides are being used, the same angle can be found using different inverse trigonometric functions.

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Exercise

Determine the measure of the angle

Show Solution
Solution

In order to solve for we must use inverse trigonometric ratios. To begin, we'll label the sides of the triangle relative to

Since the hypotenuse and the adjacent side are given, we'll use

Since, it follows that
Thus, the measure of the angle is
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