Warm-Up Exercises 6, a = 8 b 10 c = 10, c = 7 b a = In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing length. Give exact values. 1. 6, a = 8 b ANSWER 10 c = 2. 10, c = 7 b ANSWER a = 51 If you walk 2.0 kilometers due east and then 1.5 kilometers due north, how far will you be from your starting point? 3. ANSWER 2.5 km

Lesson 12.1 Right Triangle Trigonometry

Your Notes

Keywords: sine, cosine, tangent, cosecant, cotangent

Each of the six possible ratios of the three sides of a right triangle is used to define a trigonometric function.

You will use an airplane’s height and angle of descent to find this distance.

q Evaluate Trigonometric Functions Example 1 Evaluate the six trigonometric functions of the angle shown in the triangle. q SOLUTION From the Pythagorean theorem, the length of the hypotenuse is, 5 2 122 + = 169 13 Using adj = 5, opp = 12, and hyp =13, you can write the following. 9

q q q q Evaluate Trigonometric Functions Example 1 = hyp opp sin 13 12 adj cos q 13 5 = adj opp tan q 5 12 = opp hyp csc q 12 13 adj sec 5 cot 10

Checkpoint Evaluate Trigonometric Functions 1. Evaluate the six trigonometric functions of the angle shown in the triangle. q ANSWER , = sin q 25 7 cos 24 tan csc cot sec

Find a Missing Side Length Example 2 Find a Missing Side Length Find the value of x for the right triangle shown. SOLUTION Write an equation using a trigonometric function that involves the ratio of x and 8. Then solve the equation for x. hyp adj = cos 30° Write trigonometric equation. 8 x = cos 30° Substitute. 15

Find a Missing Side Length Example 2 Find a Missing Side Length 3 x = Use table to find 30º 2 8 x = 4 3 Multiply each side by 8. ANSWER The length of the longer leg is 6.93. 4 3 ≈ 16

Find an Unknown Side Length Checkpoint Find an Unknown Side Length Find the value of x for the right triangle. 2. ANSWER 4 3 3. ANSWER 9 2 4. 13 2 ANSWER

q Use a Calculator Example 3 Find the value of x for the right triangle. a. b. SOLUTION a. = adj opp tan q b. hyp cos = 10 x tan 25° 3 cos 63° 18

Example 3 Use a Calculator = x ( ) tan 25° 10 = 3 ( ) cos 63° x x 4.7 ≈ = cos 63° 3 x 6.6 ≈ 19

q Use Trigonometry in Real-life Write trigonometric equation. Example 4 Use Trigonometry in Real-life Planes How far is the plane from the landing spot? SOLUTION Write an equation using a trigonometric function that involves the ratio of x and 800. Then solve the equation for x. = opp hyp csc q Write trigonometric equation. = 800 x csc 15° Substitute. 20

Use Trigonometry in Real-life Example 4 Use Trigonometry in Real-life = 800 x sin 15° 1 Use the reciprocal of csc 15°. Multiply each side by 800. x 3091 ≈ ANSWER The plane is about 3091 feet from the landing spot. 21

Use Trigonometric Equations Checkpoint Use Trigonometric Equations 5. Write a trigonometric equation to find the length of the hypotenuse in part (a) of Example 3. ANSWER = y 10 cos 25° sec 25° or In Example 4, how far is the plane from the landing spot if the plane is flying at a height of 650 feet and is heading toward the landing spot at an angle of 20°? 6. ANSWER about 1900 ft

Suppose you are standing on the Grand View Terrace viewing platform at Mount Rushmore 550 feet from the base of the monument. You look up at the top of Mount Rushmore at an angle of 39°. How high is the top of the monument from where you are standing?

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