9.1 Use Trigonometry with Right Triangles

Transcription

1 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.

2 Pythagorean Theorem In a right triangle, the square of the length of the otenuse is equal to the sum of the squares of the lengths of the legs. a 2 + b 2 = c 2 c a b

3 In right triangle ABC, a and b are the lengths of the legs and c is the length of the otenuse. Find the missing length. Give exact values. 1. a = 6, b = 8 ANSWER c = c = 10, b = 7 ANSWER a = 51

4 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.

5 Right triangle parts. B otenuse leg A C leg Name the otenuse. Name the legs. θ

6 Opposite or Adjacent? The opposite of east is west The door is opposite the windows. In a ROY G BIV, red is acent to orange The door is acent to the white boards.

7 B Triangle Parts otenuse c a A C b Opposite Adjacent side side Opposite side Adjacent side Which side is the acent? Looking from angle A: Which side is the otenuse? Which side is the opposite? Which side is the acent? Looking from angle B: Which side is the opposite? c a b b a

8 Why do I care what side is opposite or acent? Answer: Trigonometric Ratios Sine, Cosine, Tangent, Cosecant, Secant, Cotangent

9 Trigonometric ratios B c a A b C

10 Trigonometric ratios and definition abbreviations B c a A b C SOH-CAH-TOA

11 Trigonometric ratios and definition B abbreviations c a A b C

12 Evaluate the six trigonometric functions of the angle θ. 1. SOLUTION From the Pythagorean theorem, the length of the otenuse is = 25 = 5. sin θ = opp = 3 5 csc θ = opp = 5 3 cos θ = = 4 5 sec θ = = 5 4 tan θ = opp = 3 4 cot θ = opp = 4 3

13 Evaluate the six trigonometric functions of the angle θ. SOLUTION 3. From the Pythagorean theorem, the length of the acent is sin θ = cos θ = opp = 5 csc θ = 5 2 opp = = sec θ = 5 2 = tan θ = opp = 5 5 = 1 cot θ = opp = 5 = 1 5

14 Right Triangle In a triangle, the otenuse is 2 times as long as either leg. The ratios of the side lengths can be written l-l-l 2. l l

15 Right Triangle In a triangle, the otenuse 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. l 60 2l 30

16 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 otenuse has length 10. By the Pythagorean theorem, the length x of the other leg is x = STEP 2 Find: the value of sin θ = 51. sin θ = opp = ANSWER sin θ = 51 10

17 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 = Write trigonometric equation. 3 x = 2 8 Substitute. 4 3 = x Multiply each side by 8. ANSWER The length of the side is x =

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

19 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 cos 45 = = 45º. sin 45º = opp a cos 45º = sin 45º b = 5 5 5(cos 45º) = a 5(sin45º) = b 3.54 a 3.54 b Write trigonometric equation. Substitute. Solve for the variable. Use a calculator. ANSWER So, A = 45º, b 3.54, and a 3.54.

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

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

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

23 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 solve an equation to find the height h. h sin 48º = Write trigonometric equation (sin 48º) = h Multiply each side by x Use a calculator. ANSWER The height of the parasailer above the boat is about 223 feet.

24 Use the Pythagorean Theorem to find missing lengths in right triangles. 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.

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