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9.1 Use Trigonometry with Right Triangles • Use the Pythagorean Theorem to find missing lengths in right triangles. • Find trigonometric ratios using right triangles. • Use trigonometric ratios to find angle measures in right triangles. • Use special right triangles to find lengths in right triangles. Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. c a2 + b2 = c2 a b 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. a = 6, b = 8 ANSWER c = 10 2. c = 10, b = 7 ANSWER a = 51 Definition Trigonometry—the study of the special relationships between the angle measures and the side lengths of right triangles. A trigonometric ratio is the ratio the lengths of two sides of a right triangle. Right triangle parts. B hypotenuse leg A C leg Name the hypotenuse. AB Name the legs. BC, AC The Greek letter theta Opposite or Adjacent? west • The opposite of east is ______ • The door is opposite the windows. • In a ROY G BIV, red is adjacent to orange _______ • The door is adjacent to the cabinet. Triangle Parts B hypotenuse c A Opposite side Adjacent side a C b Opposite Adjacent side side Looking from angle B: Which side is the opposite? b Which side is the adjacent? a Looking from angle A: Which side is the hypotenuse? c Which side is the opposite? a Which side is the adjacent? b Trigonometric ratios B BC a lengthof leg oppositeA sine of A AB c lengthof hypotenuse cosecant of ∠ = hypotenuse = length of leg opposite = c a A length of leg adjacent A AC b cosine of A AB c length of hypotenuse hypotenuse secant of ∠ = = = length of leg adjacent ∠ b BC a lengthof leg oppositeA tangentof A AC b lengthof leg adjacenttoA length of leg adjacent∠ cotangent of ∠ = = = length of leg opposite ∠ C Trigonometric ratios and definition abbreviations opposite a c sin A hypotenuse c adjacent b cos A hypotenuse c opposite a tan A adjacent b SOHCAHTOA A b B a C Trigonometric ratios and definition abbreviations c ℎ ∠ = s ∠ = ℎ ∠ = A 1 = −1 1 = −1 1 = −1 B a b C Evaluate the six trigonometric functions of the angle θ. 1. SOLUTION From the Pythagorean theorem, the length of the hypotenuse is √ 32 + 42 = √ 25 = 5. sin θ = opp = hyp cos θ = adj 4 = 5 hyp tan θ = opp = adj 3 5 3 4 csc θ = hyp 5 = 3 opp sec θ = hyp 5 = 4 adj cot θ = adj 4 = opp 3 Evaluate the six trigonometric functions of the angle θ. 3. SOLUTION From the Pythagorean theorem, the length of the adjacent is 5 2 2 − 52 = 25 = 5. sin θ = opp 5 = hyp 5√2 cos θ = adj 5 = hyp 5√2 tan θ = opp = adj 5 =1 5 csc θ = hyp 5√2 = 5 opp sec θ = hyp 5√2 = 5 adj cot θ = adj 5 =1 = opp 5 45°- 45°- 90° Right Triangle In a 45°- 45°- 90° triangle, the hypotenuse is √2 times as long as either leg. The ratios of the side lengths can be written l-l-l√2. l 2 l l 30°- 60° - 90° Right Triangle In a 30°- 60° - 90° triangle, the hypotenuse is twice as long as the shorter leg (the leg opposite the 30° angle, and the longer leg (opposite the 60° angle) is √3 times as long as the shorter leg. The ratios of the side lengths can be written l - l√3 – 2l. 60° 2l l 30° l 3 4. In a right triangle, θ is an acute angle and cos θ = 7 . What is sin θ? 10 SOLUTION STEP 1 Draw: a right triangle with acute angle θ such that the leg opposite θ has length 7 and the hypotenuse has length 10. By the Pythagorean theorem, the length x of the other x = √ 102 – 72 = √ 51. leg is STEP 2 Find: the value of sin θ. sin θ = opp √ 51 = 10 hyp ANSWER sin θ = √ 51 10 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 cos 30º = Write trigonometric equation. hyp √3 x = 8 2 4√ 3 = x Substitute. Multiply each side by 8. ANSWER The length of the side is x = 4 √ 3 6.93. Make sure your calculator is set to degrees. ABC. Solve SOLUTION A and B are complementary angles, so B = 90º – 28º = 68º. tan 28° = opp adj a tan 28º = 15 15(tan 28º) = a 7.98 a ANSWER cos 28º = adj hyp 15 cos 28º = c c∙ 28° = 15 15 = 28° c 17.0 So, B = 62º, a Write trigonometric equation. Substitute. Solve for the variable. Use a calculator. 7.98, and c 17.0. Solve ABC using the diagram at the right and the given measurements. 5. B = 45°, c = 5 SOLUTION A and B are complementary angles, so A = 90º – 45º = 45º. adj cos 45° = sin 45º = hyp a cos 45º = 5 5(cos 45º) = a 3.54 a ANSWER opp hyp b 5 5(sin 45º) = b sin 45º = 3.54 So, A = 45º, b b Write trigonometric equation. Substitute. Solve for the variable. Use a calculator. 3.54, and a 3.54. 6. A = 32°, b = 10 SOLUTION A and B are complementary angles, so B = 90º – 32º = 58º. opp hyp tan 32° = sec 32º = adj adj a tan 32º = 10 10(tan 32º) = a 6.25 a ANSWER sec 32º = c 10 Write trigonometric equation. Substitute. Solve for the variable. 11.8 c So, B = 58º, a Use a calculator. 6.25, and c 11.8. 7. A = 71°, c = 20 SOLUTION A and B are complementary angles, so B = 90º – 71º = 19º. adj opp cos 71° = sin 71º = hyp hyp b cos 71º = 20 20(cos 71º) = b 6.51 ANSWER b a sin 71º = 20 20(sin 71º) = a 18.9 a So, B = 19º, b Write trigonometric equation. Substitute. Solve for the variable. Use a calculator. 6.51, and a 18.9. 8. B = 60°, a = 7 SOLUTION A and B are complementary angles, so A = 90º – 60º = 30º. hyp opp Write trigonometric sec 60° = tan 60º = adj adj equation. c 7 sec 60º = ) ( 1 7 =c cos 60º 14 = c ANSWER b tan 60º = 7 7(tan 60º) = b 12.1 b Substitute. Solve for the variable. Use a calculator. So, A = 30º, c = 14, and b 12.1. 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. SOLUTION STEP 1 Draw: a diagram that represents the situation. STEP 2 Write: and h solve an equation to find the height h. sin 48º = Write trigonometric equation. 300 300(sin 48º) = h Multiply each side by 300. 223 ≈ x Use a calculator. ANSWER The height of the parasailer above the boat is about 223 feet. • Use the Pythagorean Theorem to find missing lengths in right triangles. 2 + 2 = 2 • Find trigonometric ratios using right triangles. Since, cosine, tangent, cotangent, secant, cosecant • Use trigonometric ratios to find angle measures in right triangles. • Use special right triangles to find lengths in right triangles. In a 45°-45°-90° right triangle, use , , 2 In a 30°-60°-90° right triangle, use , 2, 2 9.1 Assignment Page 560, 4-14 even, 17, 19, 22-26 even, skip 8