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EXAMPLE 1

Evaluate the six trigonometric functions of the angle θ. From the Pythagorean theorem, the length of the. hypotenuse is. √. 169. 13. √. 5 2 + 12 2. =. =. opp. hyp. 13. 12. sin θ. csc θ. =. =. =. =. 12. hyp. opp. 13. EXAMPLE 1. Evaluate trigonometric functions. SOLUTION.

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EXAMPLE 1

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  1. Evaluate the six trigonometric functions of the angle θ. From the Pythagorean theorem, the length of the hypotenuse is √ 169 13. √ 52 + 122 = = opp hyp 13 12 sinθ cscθ = = = = 12 hyp opp 13 EXAMPLE 1 Evaluate trigonometric functions SOLUTION

  2. 5 13 adj hyp cosθ secθ = = = = 13 5 hyp adj 12 5 opp adj tanθ cotθ = = = = 5 12 adj opp EXAMPLE 1 Evaluate trigonometric functions

  3. Draw: a right triangle with acute angle θ such that the leg opposite θ has length 4 and the hypotenuse has length 7. By the Pythagorean theorem, the length xof the other leg is x = 33. √ 72 – 42 √ = EXAMPLE 2 Standardized Test Practice SOLUTION STEP 1

  4. opp 4 33 4 tanθ = = = adj √ 33 33 ANSWER The correct answer is B. √ EXAMPLE 2 Standardized Test Practice Find the value of tan θ. STEP 2

  5. Find the value of x for the right triangle shown. adj cos30º = hyp √ 3 x = 8 2 EXAMPLE 3 Find an unknown side length of a right triangle SOLUTION Write an equation using a trigonometric function that involves the ratio of x and 8. Solve the equation for x. Write trigonometric equation. Substitute.

  6. x √ 4 3 = ANSWER √ 4 3 The length of the side is x= 6.93. EXAMPLE 3 Find an unknown side length of a right triangle Multiply each side by 8.

  7. Solve ABC. A and B are complementary angles, so B= 90º – 28º opp hyp tan28° sec28º = = adj adj a c tan28º sec28º = = 15 15 EXAMPLE 4 Use a calculator to solve a right triangle SOLUTION = 68º. Write trigonometric equation. Substitute.

  8. a c 7.98 17.0 So,B = 62º, a 7.98, and c 17.0. ANSWER ) ( 1 15 = c cos28º EXAMPLE 4 Use a calculator to solve a right triangle Solve for the variable. 15(tan28º) = a Use a calculator.

  9. While standing at Yavapai Point near the Grand Canyon, you measure an angle of 90º between Powell Point and Widforss Point, as shown. You then walk to Powell Point and measure an angle of 76º between Yavapai Point and Widforss Point. The distance between Yavapai Point and Powell Point is about 2 miles. How wide is the Grand Canyon between Yavapai Point and Widforss Point? EXAMPLE 5 Use indirect measurement Grand Canyon

  10. x tan 76º = 2 x 8.0 ≈ ANSWER The width is about 8.0 miles. EXAMPLE 5 Use indirect measurement SOLUTION Write trigonometric equation. 2(tan 76º) = x Multiply each side by 2. Use a calculator.

  11. A parasailer is attached to a boat with a rope 300 feet long. The angle of elevation from the boat to the parasailer is 48º. Estimate the parasailer’s height above the boat. EXAMPLE 6 Use an angle of elevation Parasailing

  12. ANSWER The height of the parasailer above the boat is about 223 feet. h sin 48º = 300 x 223 ≈ EXAMPLE 6 Use an angle of elevation SOLUTION STEP 1 Draw: a diagram that represents the situation. STEP 2 Write: and solve an equation to find the height h. Write trigonometric equation. 300(sin 48º) = h Multiply each side by 300. Use a calculator.

  13. a. Because 240º is 60º more than 180º, the terminal side is 60º counterclockwise past the negative x-axis. EXAMPLE 1 Draw angles in standard position Draw an angle with the given measure in standard position. a.240º SOLUTION

  14. b. Because 500º is 140º more than 360º, the terminal side makes one whole revolution counterclockwise plus 140º more. EXAMPLE 1 Draw angles in standard position Draw an angle with the given measure in standard position. b. 500º SOLUTION

  15. c. Because –50º is negative, the terminal side is 50º clockwise from the positive x-axis. EXAMPLE 1 Draw angles in standard position Draw an angle with the given measure in standard position. c. –50º SOLUTION

  16. There are many such angles, depending on what multiple of 360º is added or subtracted. EXAMPLE 2 Find coterminal angles Find one positive angle and one negative angle that are coterminal with (a) –45º and (b) 395º. SOLUTION a. –45º + 360º = 315º –45º – 360º = – 405º

  17. = –325º EXAMPLE 2 Find coterminal angles b. 395º – 360º = 35º 395º – 2(360º)

  18. π Convert (a) 125º to radians and (b) – radians to degrees. 12 ) ( πradians = 125º 180º 25π radians = 36 ( ) ( ) π π 180º – – b. radians = 12 12 π radians –15º = EXAMPLE 3 Convert between degrees and radians a. 125º

  19. A softball field forms a sector with the dimensions shown. Find the length of the outfield fence and the area of the field. EXAMPLE 4 Solve a multi-step problem Softball

  20. ) ( π πradians radians 90º = 90º = 2 180º EXAMPLE 4 Solve a multi-step problem SOLUTION STEP 1 Convert the measure of the central angle to radians.

  21. π ) ( Arc length: s=r= 180 = 90π ≈ 283 feet θ 2 π 1 1 ) ( Area: A=r2θ= (180)2 = 8100π ≈ 25,400 ft2 2 2 2 ANSWER The length of the outfield fence is about 283feet. The area of the field is about 25,400square feet. EXAMPLE 4 Solve a multi-step problem STEP 2 Find the arc length and the area of the sector.

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