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EXAMPLE 1 Evaluate trigonometric functions Evaluate the six trigonometric functions of the angle θ. SOLUTION From the Pythagorean theorem, the length of the hypotenuse is √ 52 + 122 = √ 169 = 13. sin θ = opp 12 = hyp 13 csc θ = hyp 13 = 12 opp EXAMPLE 1 Evaluate trigonometric functions cos θ = adj = hyp 5 13 tan θ = opp = adj 12 5 sec θ = 13 hyp = 5 adj cot θ = 5 adj = 12 opp EXAMPLE 2 Standardized Test Practice SOLUTION STEP 1 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 x of the other leg is x = √ 72 – 42 = √ 33. EXAMPLE 2 Standardized Test Practice STEP 2 Find the value of tan θ. tan θ = 4 opp = adj √ 33 4 √ 33 = 33 ANSWER The correct answer is B. EXAMPLE 3 Find an unknown side length of a right triangle 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. Solve the equation for x. adj Write trigonometric equation. cos 30º = hyp √3 2 = x 8 Substitute. EXAMPLE 3 4√3 = x Find an unknown side length of a right triangle Multiply each side by 8. ANSWER The length of the side is x = 4 √ 3 6.93. EXAMPLE 4 Solve Use a calculator to solve a right triangle ABC. SOLUTION A and B are complementary angles, so B = 90º – 28º = 68º. opp tan 28° = adj a tan 28º = 15 sec 28º = hyp adj c sec 28º = 15 Write trigonometric equation. Substitute. EXAMPLE 4 15(tan 28º) = a 7.98 a Use a calculator to solve a right triangle 15 ( 1 cos 28º 17.0 )=c c ANSWER So, B = 62º, a 7.98, and c 17.0. Solve for the variable. Use a calculator. EXAMPLE 5 Use indirect measurement Grand Canyon 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 SOLUTION tan 76º = x 2 2(tan 76º) = x 8.0 ≈ x ANSWER The width is about 8.0 miles. Write trigonometric equation. Multiply each side by 2. Use a calculator. EXAMPLE 6 Use an angle of elevation Parasailing 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 SOLUTION STEP 1 Draw: a diagram that represents the situation. STEP 2 Write: and solve an equation to find the height h. h Write trigonometric equation. sin 48º = 300 300(sin 48º) = h Multiply each side by 300. Use a calculator. 223 ≈ x ANSWER The height of the parasailer above the boat is about 223 feet. EXAMPLE 1 Draw angles in standard position Draw an angle with the given measure in standard position. a. 240º SOLUTION 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. b. 500º SOLUTION 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. c. –50º SOLUTION c. Because –50º is negative, the terminal side is 50º clockwise from the positive x-axis. EXAMPLE 2 Find coterminal angles Find one positive angle and one negative angle that are coterminal with (a) –45º and (b) 395º. SOLUTION There are many such angles, depending on what multiple of 360º is added or subtracted. a. –45º + 360º = 315º –45º – 360º = – 405º EXAMPLE 2 Find coterminal angles b. 395º – 360º = 35º 395º – 2(360º) = –325º EXAMPLE 3 Convert between degrees and radians Convert (a) 125º to radians and (b) – degrees. π radians a. 125º = 125º 180º ) ( 25π 36 = π b. – 12 = radians π – radians 12 ( = –15º π radians to 12 )( 180º π radians ) EXAMPLE 4 Solve a multi-step problem Softball 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 SOLUTION STEP 1 Convert the measure of the central angle to radians. 90º = 90º ( π radians 180º )= π 2 radians EXAMPLE 4 Solve a multi-step problem STEP 2 Find the arc length and the area of the sector. Arc length: s = r θ = 180 ( π ) = 90π ≈ 283 feet 2 Area: A = 25,400 ft2 1 2 rθ= 2 1 2 (180) 2 π ( 2 )= 8100π ≈ ANSWER The length of the outfield fence is about 283 feet. The area of the field is about 25,400 square feet.