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9.1 Use Trigonometry with Right Triangles

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Presentation on theme: "9.1 Use Trigonometry with Right Triangles"β€” Presentation transcript:

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 hypotenuse is equal to the sum of the squares of the lengths of the legs. a2 + b2 = c2 a c b

3 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 =

4 Definition Trigonometryβ€”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. Name the hypotenuse. Name the legs.
B C hypotenuse leg leg Name the hypotenuse. Name the legs. The Greek letter theta 𝜽

6 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

7 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

8 Trigonometric ratios A B C a b c cosecant of ∠𝐴= hypotenuse length of leg opposite = 𝐴𝐡 𝐡𝐢 = 𝑐 π‘Ž secant of ∠𝐴= hypotenuse length of leg adjacent ∠𝐴 = 𝐴𝐡 𝐴𝐢 = 𝑐 𝑏 cotangent of ∠𝐴= length of the hypotenuse length of leg opposite ∠𝐴 = 𝐴𝐡 𝐡𝐢 = 𝑐 π‘Ž

9 Trigonometric ratios and definition abbreviations
SOHCAHTOA

10 Trigonometric ratios and definition abbreviations
𝑐𝑠𝑐= 1 𝑠𝑖𝑛 𝑐𝑠𝑐 ∠𝐴= β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ 𝑠𝑖𝑛 βˆ’1 𝑠𝑒𝑐= 1 π‘π‘œπ‘  s𝑒𝑐 ∠𝐴= β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘π‘œπ‘  βˆ’1 π‘π‘œπ‘‘βˆ π΄= π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’ π‘π‘œπ‘‘= 1 π‘‘π‘Žπ‘› π‘‘π‘Žπ‘› βˆ’1

11 Evaluate the six trigonometric functions of the angle ΞΈ.
1. SOLUTION From the Pythagorean theorem, the length of the hypotenuse is √ 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

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

13 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

14 30Β°- 60Β° - 90Β° Right Triangle In a 30Β°- 60Β° - 90Β° triangle, the hypotenuse is twice as long as the shorter leg (the leg opposite the 30Β° angle, and the longer leg (opposite the 60Β° angle) is √3 tunes as long as the shorter leg. The ratios of the side lengths can be written l - l√3 – 2l. 60Β° 30Β° l 2l

15 4. In a right triangle, ΞΈ is an acute angle and cos ΞΈ = What is sin ΞΈ? 7 10 SOLUTION STEP 1 Draw: a right triangle with acute angle ΞΈ such that the leg opposite ΞΈ 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 ΞΈ. sin ΞΈ = opp hyp = 51 √ 10 ANSWER sin ΞΈ = 51 √ 10

16 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

17 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 = 90ΒΊ – 28ΒΊ = 68ΒΊ. tan 28Β° = opp adj cos 28ΒΊ = adj hyp Write trigonometric equation. tan 28ΒΊ = a 15 cos 28ΒΊ = 15 c Substitute. cβˆ™π‘π‘œπ‘ 28Β°=15 Solve for the variable. 15(tan 28ΒΊ) = a 𝑐= 15 π‘π‘œπ‘ 28Β° Use a calculator. 7.98 a 17.0 c So, B = 62ΒΊ, a , and c ANSWER

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

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

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

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

22 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 48ΒΊ. 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 48ΒΊ = h 300 Write trigonometric equation. 300(sin 48ΒΊ) = h Multiply each side by 300. 223 β‰ˆ x Use a calculator. The height of the parasailer above the boat is about 223 feet. ANSWER

23 Use the Pythagorean Theorem to find missing lengths in right triangles.
π‘Ž 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 45Β°-45Β°-90Β° right triangle, use 𝑙, 𝑙, 𝑙 2 In a 30Β°-60Β°-90Β° right triangle, use 𝑙, 𝑙 2, 2𝑙

24 9.1 Assignment Page 560, even, 17, 19, even, skip 8


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