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Example 1

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. 12 2. 169. 13. =. =. +. Using adj 5 , opp 12 , and hyp 13 , you can write the following. =. =. =.

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Example 1

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  1. 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, 52 122 169 13 = = + Using adj5, opp12, and hyp13, you can write the following. = = = Evaluate Trigonometric Functions

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

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

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

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

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

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

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

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

  10. Example 4 1 x = 800 sin 15° Multiply each side by 800. 3091 x ≈ ANSWER The plane is about 3091 feet from the landing spot. Use Trigonometry in Real-life Use the reciprocal of csc15°.

  11. Checkpoint y 10 ANSWER cos 25° or sec 25° = = y 10 6. 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°? about 1900 ft ANSWER Use Trigonometric Equations Write a trigonometric equation to find the length of the hypotenuse in part (a) of Example 3. 5.

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