Warm Up

Lesson Presentation

Lesson Quiz

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 ° 3. 24 ° 2. 60 ° 4. 38 °

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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

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13-1 Right-Angle Trigonometry

Objectives

Understand and use trigonometric relationships of acute angles in triangles.

Determine side lengths of right triangles by using trigonometric functions.

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13-1 Right-Angle Trigonometry

Vocabulary

trigonometric function sine cosine tangent cosecants secant cotangent

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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.

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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.

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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 =

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13-1 Right-Angle Trigonometry Check It Out!

Example 1 Find the value of the sine, cosine, and tangent functions for θ.

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13-1 Right-Angle Trigonometry

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

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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.

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

Substitute for sin 30 °.

Multiply both sides by 74 to solve for x.

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13-1 Right-Angle Trigonometry Check It Out!

Example 2 Use a trigonometric function to find the value of x.

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13-1 Right-Angle Trigonometry Example 3: Sports Application In 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?

Substitute 15.1

° for θ, h for opp., and 19 for hyp.

Multiply both sides by 19.

5 ≈ h

Use a calculator to simplify.

The height above the water is about 5 ft.

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13-1 Right-Angle Trigonometry Caution!

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.

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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?

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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.

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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.

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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 144 ≈ x

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

Multiply both sides by 180.

Use a calculator to solve for x.

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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.

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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

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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.

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13-1 Right-Angle Trigonometry Check It Out!

Example 4 Continued

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

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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.

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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.

a

2 + b 2 = c 2

c

2 = 24 2 + 70 2

c

2 = 5476

Pythagorean Theorem.

Substitute 24 for a and 70 for b.

Simplify.

70

c = 74

Solve for c. Eliminate the negative solution.

24

θ

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13-1 Right-Angle Trigonometry Example 5 Continued

Step 2 Find the function values.

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13-1 Right-Angle Trigonometry Helpful Hint

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

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13-1 Right-Angle Trigonometry Check It Out!

Example 5 Find the values of the six trigonometric functions for θ. 80 18

θ

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