Section 10.6 Right Triangle Trigonometry Math in Our World

Learning Objectives Find basic trigonometric ratios. Use trigonometric ratios to find sides of a right triangle. Use trigonometric ratios to find angles of a right triangle. Solve problems using trigonometric ratios.

Trigonometry In this section we will study the basics of trigonometry, a very old subject whose name literally means “angle measurement.” In trigonometry, we use relationships among the sides and angles of triangles to solve problems. The big restriction we will work with when studying the basics of trigonometry is that all triangles have to be right triangles.

Trigonometry In this section, we will use capital letters A, B, and C to represent the angles, and lowercase letters a, b, and c to represent the lengths of the sides. We will always use C to represent the right angle. In each case, the letter of an angle matches the letter of the side across from it.

Trigonometry There are three basic trigonometric ratios. They are called the sine (abbreviated sin), the cosine (abbreviated cos), and the tangent (abbreviated tan).

Trigonometry There are three basic trigonometric ratios. They are called the sine (abbreviated sin), the cosine (abbreviated cos), and the tangent (abbreviated tan).

Trigonometry There are three basic trigonometric ratios. They are called the sine (abbreviated sin), the cosine (abbreviated cos), and the tangent (abbreviated tan).

EXAMPLE 1 Finding Basic Trigonometric Ratios For the triangle shown , find sin B, cos B, and tan B. Since two of the three ratios involve the length of the hypotenuse, we need to find that first, using the Pythagorean Theorem. SOLUTION

EXAMPLE 1 Finding Basic Trigonometric Ratios Now we can use the trigonometric ratios from the definitions above; the hypotenuse is 5 m, the side opposite B is 4 m, and the side adjacent to B is 3 m. SOLUTION

Trigonometry Here we see two different right triangles with a 30° angle. Because the triangles are similar, the ratios of corresponding sides are equal, so sin 30 is 1/2 in both triangles. One situation that trigonometry is often used for is to find the lengths of sides of a triangle. If we know one side and one of the acute angles, we can find either of the remaining sides.

EXAMPLE 2 Finding a Side of a Triangle Using Tangent In the right triangle ABC, find the length of side a when m∠A = 30° and b = 200 feet. SOLUTION Step 1 Draw and label the figure. Step 2 Choose an appropriate trigonometric ratio and substitute values into it. In this case, we know the side adjacent to angle A, and are asked to find the side opposite, so tangent is a good choice.

EXAMPLE 2 Finding a Side of a Triangle Using Tangent SOLUTION Step 3 Solve the resulting equation for a. Multiply both sides by 200. To evaluate this answer, we use a calculator in degree mode and press 200 x 30 Tan = (standard scientific calculator), or 200 Tan 30 Enter (standard graphing calculator). The result in either case, rounded to two decimal places, is 115.47, so the length of side a is 115.47 feet.

EXAMPLE 3 Finding a Side of a Triangle Using Cosine In the right triangle ABC, find the measure of side c when m∠B = 72 and the length of side a is 24 ft. SOLUTION Step 1 Draw and label the figure. Step 2 We know the side adjacent to B and want to find the hypotenuse, so cosine is a good choice.

EXAMPLE 3 Finding a Side of a Triangle Using Cosine SOLUTION Step 3 Solve for c. Multiply both sides by c. Divide both sides by cos 72°.

Trigonometry Trigonometric ratios can be used to find angles as well, provided that we know at least two sides of a right triangle. To accomplish this, we will need to use special routines programmed into a calculator known as inverse trigonometric functions. They are accessed using the 2nd key on most calculators.

EXAMPLE 4 Finding an Angle Using Trigonometric Ratios In the right triangle ABC, side c (the hypotenuse) measures 25 inches and side b measures 24 inches. Find the measure of angle B. SOLUTION Step 1 Draw and label the figure. Step 2 We know the hypotenuse and the side opposite angle B, so sine is a good choice.

EXAMPLE 4 Finding an Angle Using Trigonometric Ratios SOLUTION Step 3 Solve for B. To accomplish this, we will need to access the inverse sine feature on a calculator: .96 2nd Sin (standard scientific calculator), or 2nd Sin .96 Enter (standard graphing calculator). The result, rounded to two decimal places, is 73.74, so m∠B = 73.74°.

EXAMPLE 5 An Application of Trigonometry to Home Improvement The safety label on a 12-foot ladder says that the ladder should not be placed at an angle steeper than 65 with the ground. What is the closest safe distance between the base of the ladder and the wall?

EXAMPLE 5 An Application of Trigonometry to Home Improvement SOLUTION Step 3 Solve for b. The bottom of the ladder can’t be any closer than 5.07 feet from the wall. Step 1 Draw and label the figure. Step 2 Choose an appropriate trigonometric ratio and substitute values into it. Since we need to find the length of side b and we are given the measures of angle A and side c, cosine is a good choice.

Elevation and Depression The angle of elevation of an object is the measure of the angle from a horizontal line at the point of an observer upward to the line of sight to the object. The angle of depression is the measure of an angle from a horizontal line at the point of an observer downward to the line of sight to the object.

EXAMPLE 6 Finding the Height of a Tall Object Using Trigonometry In order to find the height of a building, an observer who is 6 feet tall measures the angle of elevation from his head to the top of the building to be 32 when he is standing at a point 200 feet from the building. How tall is the building? SOLUTION Step 1 Draw and label the figure.

We see that the height of the building is side a (125 feet) plus 6 feet, so the building is 131 feet tall. EXAMPLE 6 Finding the Height of a Tall Object Using Trigonometry SOLUTION Step 2 Choose an appropriate trigonometric ratio and substitute values into it. Since we are given the measure of angle A and the measure of side b, we can use tangent. Step 3 Solve for a.

EXAMPLE 7 Finding an Angle of Depression A local photographer gets a hot tip that a famous starlet is sunbathing on a boat cruising down a river. He sets up his camera on a bridge that is 22 feet above the water and 60 feet from a bend in the river, hoping to get a shot as the boat comes into view. At what angle of depression should he aim the camera?

EXAMPLE 7 Finding an Angle of Depression SOLUTION The angle of depression should be a bit less than 70°. Step 1 Draw and label the figure. Step 2 We are looking for angle B, and have the opposite and adjacent sides, so we choose tangent. Step 3 Solve for B using the inverse tangent feature on a calculator.