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. a2 + b2 = c2 a c 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. Name the hypotenuse. Name the legs. B C hypotenuse leg leg Name the hypotenuse. Name the legs. The Greek letter theta 𝜽

Opposite or Adjacent? west The opposite of east is ______ The door is opposite the windows. In a ROY G BIV, red is adjacent to _______ The door is adjacent to the cabinet. orange

Triangle Parts Looking from angle B: Which side is the opposite? Opposite side hypotenuse Adjacent side c Looking from angle B: Which side is the opposite? Which side is the adjacent? a b a b Opposite side Adjacent side Looking from angle A: Which side is the hypotenuse? Which side is the opposite? Which side is the adjacent? c a b

Trigonometric ratios A B C a b c cosecant of ∠𝐴= hypotenuse length of leg opposite = 𝐴𝐵 𝐵𝐶 = 𝑐 𝑎 secant of ∠𝐴= hypotenuse length of leg adjacent ∠𝐴 = 𝐴𝐵 𝐴𝐶 = 𝑐 𝑏 cotangent of ∠𝐴= length of the hypotenuse length of leg opposite ∠𝐴 = 𝐴𝐵 𝐵𝐶 = 𝑐 𝑎

Trigonometric ratios and definition abbreviations SOHCAHTOA

Trigonometric ratios and definition abbreviations 𝑐𝑠𝑐= 1 𝑠𝑖𝑛 𝑐𝑠𝑐 ∠𝐴= ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑛 −1 𝑠𝑒𝑐= 1 𝑐𝑜𝑠 s𝑒𝑐 ∠𝐴= ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑐𝑜𝑠 −1 𝑐𝑜𝑡∠𝐴= 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑐𝑜𝑡= 1 𝑡𝑎𝑛 𝑡𝑎𝑛 −1

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 = 3 5 csc θ = hyp opp = 5 3 cos θ = adj hyp = 4 5 sec θ = hyp adj = 5 4 tan θ = opp adj = 3 4 cot θ = adj opp = 4 3

Evaluate the six trigonometric functions of the angle θ. SOLUTION From the Pythagorean theorem, the length of the adjacent is 5 2 2 − 5 2 = 25 =5. sin θ = opp hyp = 5 5 2 √ csc θ = hyp opp 5 = 5 2 √ cos θ = adj hyp 5 = 5 2 √ sec θ = hyp adj 5 = 5 2 √ tan θ = opp adj = 5 cot θ = adj opp = 5 = 1 = 1

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 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 tunes as long as the shorter leg. The ratios of the side lengths can be written l - l√3 – 2l. 60° 30° l 2l

4. In a right triangle, θ is an acute angle and cos θ = . What is sin θ? 7 10 SOLUTION STEP 1 Draw: a right triangle with acute angle θ such that the leg opposite θ has length 10 and the hypotenuse has length 7. By the Pythagorean theorem, the length x of the other leg is 102 – 72 √ x = 51. = √ STEP 2 Find: the value of sin θ. sin θ = opp hyp = 51 √ 10 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. cos 30º = adj hyp Write trigonometric equation. 3 2 √ = x 8 Substitute. 3 4 √ = x Multiply each side by 8. The length of the side is x = 3 4 √ 6.93. ANSWER

Make sure your calculator is set to degrees. Solve ABC. Make sure your calculator is set to degrees. SOLUTION A and B are complementary angles, so B = 90º – 28º = 68º. tan 28° = opp adj cos 28º = adj hyp Write trigonometric equation. tan 28º = a 15 cos 28º = 15 c Substitute. c∙𝑐𝑜𝑠28°=15 Solve for the variable. 15(tan 28º) = a 𝑐= 15 𝑐𝑜𝑠28° Use a calculator. 7.98 a 17.0 c So, B = 62º, a 7.98, and c 17.0. ANSWER

5. B = 45°, c = 5 Solve ABC using the diagram at the right and the given measurements. SOLUTION A and B are complementary angles, so A = 90º – 45º = 45º. cos 45° = adj hyp sin 45º = opp hyp Write trigonometric equation. cos 45º = a 5 sin 45º = 5 b Substitute. 5(cos 45º) = a 5(sin 45º) = b Solve for the variable. 3.54 b Use a calculator. 3.54 a So, A = 45º, b 3.54, and a 3.54. ANSWER

6. A = 32°, b = 10 SOLUTION A and B are complementary angles, so B = 90º – 32º = 58º. tan 32° = opp adj sec 32º = hyp adj Write trigonometric equation. tan 32º = a 10 sec 32º = 10 c Substitute. 10(tan 32º) = a Solve for the variable. 11.8 c Use a calculator. 6.25 a So, B = 58º, a 6.25, and c 11.8. ANSWER

7. A = 71°, c = 20 SOLUTION A and B are complementary angles, so B = 90º – 71º = 19º. cos 71° = adj hyp sin 71º = opp hyp Write trigonometric equation. cos 71º = b 20 sin 71º = a 20 Substitute. Solve for the variable. 20(cos 71º) = b 20(sin 71º) = a Use a calculator. 6.51 b 18.9 a So, B = 19º, b 6.51, and a 18.9. ANSWER

8. B = 60°, a = 7 SOLUTION A and B are complementary angles, so A = 90º – 60º = 30º. sec 60° = hyp adj tan 60º = opp adj Write trigonometric equation. sec 60º = 7 c tan 60º = b 7 Substitute. 7 1 ( cos 60º ) = c 7(tan 60º) = b Solve for the variable. 12.1 b Use a calculator. 14 = c So, A = 30º, c = 14, and b 12.1. ANSWER

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

Use the Pythagorean Theorem to find missing lengths in right triangles. 𝑎 2 + 𝑏 2 = 𝑐 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 𝑙, 𝑙, 𝑙 2 In a 30°-60°-90° right triangle, use 𝑙, 𝑙 2, 2𝑙

9.1 Assignment Page 560, 4-14 even, 17, 19, 22-26 even, skip 8