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Trigonometry Warm Up: SOH-CAH-TOA, Missing Angles & Lengths

Learn about SOH-CAH-TOA, find missing angles and lengths in right triangles, and solve real-world applications using trigonometric functions.

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Trigonometry Warm Up: SOH-CAH-TOA, Missing Angles & Lengths

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  1. Warm Up • What does Chief “SOH-CAH-TOA” mean to you? • What is the value of the missing angle? • What is the missing length for a right triangle with leg a = 16 and hyupotenuse c = 20?

  2. What do you know about right triangles? θ

  3. Find the exact trig values for the angle x 8 x ɵ 7

  4. Check It Out! Example 1 Find the value of the sine, cosine, and tangent functions for θ. sin θ = cos θ = tan θ =

  5. Caution! Make sure that your graphing calculator is set to interpret angle values as degrees. Press . Check that Degree and not Radian is highlighted in the third row.

  6. Ex 1: Find x using sine, cosine or tangent equation • ɵ x 32° 5

  7. Ex 2: Find x using sine, cosine or tangent equation • ɵ x 23° 32

  8. Ex 3: Find x using sine, cosine or tangent equation x x 5 13

  9. Example 3: Sports Application In a waterskiing competition, a jump ramp has the measurements shown. To the nearest foot, what is the height h above water that a skier leaves the ramp? Substitute 15.1° for θ, h for opp., and 19 for hyp. Multiply both sides by 19. Use a calculator to simplify. 5 ≈ h The height above the water is about 5 ft.

  10. Check It Out! Example 3 A skateboard ramp will have a height of 12 in., and the angle between the ramp and the ground will be 17°. To the nearest inch, what will be the length l of the ramp? Substitute 17° for θ, lfor hyp., and 12 for opp. Multiply both sides by l and divide by sin 17°. Use a calculator to simplify. l ≈ 41 The length of the ramp is about 41 in.

  11. When an object is above or below another object, you can find distances indirectly by using the angle of elevation or the angle of depression between the objects.

  12. 60.7° 120 ft Check It Out! Example 4 A surveyor whose eye level is 6 ft above the ground measures the angle of elevation to the top of the highest hill on a roller coaster to be 60.7°. If the surveyor is standing 120 ft from the hill’s base, what is the height of the hill to the nearest foot? Step 1 Draw and label a diagram to represent the information given in the problem.

  13. Check It Out! Example 4 Continued Step 2 Let x represent the height of the hill compared with the surveyor’s eye level. Determine the value of x. Use the tangent function. Substitute 60.7 for θ, x for opp., and 120 for adj. 120(tan 60.7°) = x Multiply both sides by 120. 214 ≈ x Use a calculator to solve for x.

  14. Check It Out! Example 4 Continued Step 3 Determine the overall height of the roller coaster hill. x + 6 = 214 + 6 = 220 The height of the hill is about 220 ft.

  15. 80 θ 18 Check It Out! Example 5 Find the values of the six trigonometric functions for θ. Step 1 Find the length of the hypotenuse. Pythagorean Theorem. a2 + b2 = c2 Substitute 18 for a and 80 for b. c2 = 182 + 802 c2 = 6724 Simplify. Solve for c. Eliminate the negative solution. c= 82

  16. Check It Out! Example 5 Continued Step 2 Find the function values.

  17. Lesson Quiz: Part I Solve each equation. Check your answer. 1. Find the values of the six trigonometric functions for θ.

  18. Lesson Quiz: Part II 2. Use a trigonometric function to find the value of x. 3. A helicopter’s altitude is 4500 ft, and a plane’s altitude is 12,000 ft. If the angle of depression from the plane to the helicopter is 27.6°, what is the distance between the two, to the nearest hundred feet? 16,200 ft

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