Holt Algebra 2

13-1 Right-Angle Trigonometry13-1 Right-Angle Trigonometry

Holt Algebra 2

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Algebra 2

13-1 Right-Angle Trigonometry

Warm Up

Given the measure of one of the acute angles in a right triangle, find the measure of the other acute angle.

1. 45° 2. 60° 3. 24° 4. 38°

Holt Algebra 2

13-1 Right-Angle Trigonometry

Warm Up Continued

Find the unknown length for each right triangle with legs a and b and hypotenuse c.

5. b = 12, c =13 6. a = 3, b = 3

Holt Algebra 2

13-1 Right-Angle Trigonometry

Understand and use trigonometric relationships of acute angles in triangles.

Determine side lengths of right triangles by using trigonometric functions.

Objectives

Holt Algebra 2

13-1 Right-Angle Trigonometry

trigonometric functionsinecosinetangentcosecantssecantcotangent

Vocabulary

Holt Algebra 2

13-1 Right-Angle Trigonometry

A trigonometric function is a function whose rule is given by a trigonometric ratio. A trigonometric ratio compares the lengths of two sides of a right triangle. The Greek letter theta θ is traditionally used to represent the measure of an acute angle in a right triangle. The values of trigonometric ratios depend upon θ.

Holt Algebra 2

13-1 Right-Angle Trigonometry

Holt Algebra 2

13-1 Right-Angle Trigonometry

The triangle shown at right is similar to the one in the table because their corresponding angles are congruent. No matter which triangle is used, the value of sin θ is the same. The values of the sine and other trigonometric functions depend only on angle θ and not on the size of the triangle.

Holt Algebra 2

13-1 Right-Angle Trigonometry

Example 1: Finding Trigonometric Ratios

Find the value of the sine, cosine, and tangent functions for θ.

sin θ =

cos θ =

tan θ =

Holt Algebra 2

13-1 Right-Angle Trigonometry

Check It Out! Example 1

Find the value of the sine, cosine, and tangent functions for θ.

Holt Algebra 2

13-1 Right-Angle Trigonometry

You will frequently need to determine the value of trigonometric ratios for 30°,60°, and 45° angles as you solve trigonometry problems. Recall from geometry that in a 30°-60°-90° triangle, the ratio of the side lengths is 1: 3 :2, and that in a 45°-45°-90° triangle, the ratio of the side lengths is 1:1: 2.

Holt Algebra 2

13-1 Right-Angle Trigonometry

Holt Algebra 2

13-1 Right-Angle Trigonometry

Example 2: Finding Side Lengths of Special Right Triangles

Use a trigonometric function to find the value of x.

°

x = 37

The sine function relates the opposite leg and the hypotenuse.

Multiply both sides by 74 to solve for x.

Substitute for sin 30°.

Substitute 30° for θ, x for opp, and 74 for hyp.

Holt Algebra 2

13-1 Right-Angle Trigonometry

Check It Out! Example 2

Use a trigonometric function to find the value of x.

Holt Algebra 2

13-1 Right-Angle Trigonometry

Example 3: Sports ApplicationIn a waterskiing competition, a jump ramp has the measurements shown. To the nearest foot, what is the height h above water that a skier leaves the ramp?

5 ≈ h

The height above the water is about 5 ft.

Substitute 15.1° for θ, h for opp., and 19 for hyp.

Multiply both sides by 19.

Use a calculator to simplify.

Holt Algebra 2

13-1 Right-Angle Trigonometry

Make sure that your graphing calculator is set to interpret angle values as degrees. Press . Check that Degree and not Radian is highlighted in the third row.

Caution!

Holt Algebra 2

13-1 Right-Angle Trigonometry

Check It Out! Example 3 A skateboard ramp will have a height of 12 in., and the angle between the ramp and the ground will be 17°. To the nearest inch, what will be the length l of the ramp?

Holt Algebra 2

13-1 Right-Angle Trigonometry

When an object is above or below another object, you can find distances indirectly by using the angle of elevation or the angle of depression between the objects.

Holt Algebra 2

13-1 Right-Angle Trigonometry

Example 4: Geology Application

A biologist whose eye level is 6 ft above the ground measures the angle of elevation to the top of a tree to be 38.7°. If the biologist is standing 180 ft from the tree’s base, what is the height of the tree to the nearest foot?

Step 1 Draw and label a diagram to represent the information given in the problem.

Holt Algebra 2

13-1 Right-Angle Trigonometry

Example 4 Continued

Step 2 Let x represent the height of the tree compared with the biologist’s eye level. Determine the value of x.

Use the tangent function.

180(tan 38.7°) = x

Substitute 38.7 for θ, x for opp., and 180 for adj.

Multiply both sides by 180.

144 ≈ x Use a calculator to solve for x.

Holt Algebra 2

13-1 Right-Angle Trigonometry

Example 4 Continued

Step 3 Determine the overall height of the tree.

x + 6 = 144 + 6

= 150

The height of the tree is about 150 ft.

Holt Algebra 2

13-1 Right-Angle Trigonometry

Check It Out! Example 4

A surveyor whose eye level is 6 ft above the ground measures the angle of elevation to the top of the highest hill on a roller coaster to be 60.7°. If the surveyor is standing 120 ft from the hill’s base, what is the height of the hill to the nearest foot?

120 ft

60.7°

Holt Algebra 2

13-1 Right-Angle Trigonometry

Check It Out! Example 4 Continued

Step 2 Let x represent the height of the hill compared with the surveyor’s eye level. Determine the value of x.

Holt Algebra 2

13-1 Right-Angle Trigonometry

Check It Out! Example 4 Continued

Step 3 Determine the overall height of the roller coaster hill.

Holt Algebra 2

13-1 Right-Angle Trigonometry

The reciprocals of the sine, cosine, and tangent ratios are also trigonometric ratios. They are trigonometric functions, cosecant, secant, and cotangent.

Holt Algebra 2

13-1 Right-Angle Trigonometry

Example 5: Finding All Trigonometric Functions

Find the values of the six trigonometric functions for θ.

Step 1 Find the length of the hypotenuse.

70

24

θ

a2 + b2 = c2

c2 = 242 + 702

c2 = 5476

c = 74

Pythagorean Theorem.

Substitute 24 for a and 70 for b.

Simplify.

Solve for c. Eliminate the negative solution.

Holt Algebra 2

13-1 Right-Angle Trigonometry

Example 5 Continued

Step 2 Find the function values.

Holt Algebra 2

13-1 Right-Angle Trigonometry

In each reciprocal pair of trigonometric functions, there is exactly one “co”

Helpful Hint

Holt Algebra 2

13-1 Right-Angle Trigonometry

Find the values of the six trigonometric functions for θ.

80

18

θ

Check It Out! Example 5