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Geometry CP Unit 6 Lesson 6 Name ______________________ Inverse Trig Functions You know how to find the missing side lengths in a right triangle given an acute angle and the length of any side. But what if you want to find the measure of an angle? 1. Mr. Gow needs to build a wheelchair access ramp for the school’s auditorium. The ramp must rise a total of 3 feet to get from the ground to the entrance of the building. In order to follow the state building code, the angle formed by the ramp and the ground cannot exceed 4.76°. Mr. Gow has plans from the planning department that call for the ramp to start 25 feet away from the building. Will this ramp meet the state building code? a. Draw a diagram of the situation. Include as many measurements as you can. What does Mr. Gow need to find? b. Write an equation using a trig ratio. Can you solve your equation? c. To find an angle from a trig ratio you need to “undo” it. You can undo addition with subtraction, multiplication with division, or squaring by taking the square root. These examples are all pairs of inverse operations. Your calculator has a button for the inverse trigonometric functions, 𝐬𝐢𝐧!𝟏 , 𝐜𝐨𝐬 !𝟏 , and 𝐭𝐚𝐧!𝟏 . !
Check that your calculator can find an inverse trig value by typing in cos !! . Do you get 60°? !"
d. Now can you solve your equation from part (b)? According to the plan, what angle will the ramp make with the ground? Will the ramp be to code? 2. For the triangle below, find the measure of ∠𝐴 and ∠𝐵. Do you have to use trig functions to find both angles? 3. Use two different trig ratios to find the measure of ∠𝐴. Did you get the same answer both ways? 4. Calculator: find each angle measure to the nearest degree. a. sin 𝐵 = 0.4848 b. sin 𝐴 = 0.5150 c. cos 𝐴 = 0.7431 e. cos 𝜃 = 0.5878 f. tan 𝑊 = 19.0811 g. cos 𝐵 = 0.4226 d. cos 𝑊 = 0.6157 h. tan 𝜃 = 0.5317 5. Write an equation and find the measure of the angle marked with ‘?’ to the nearest degree. a. b. c. e. d. f. 6. AJ cannot figure out what he did wrong. He wrote the equation below to find the missing angle of a triangle. However, his calculator gives him an error message each time he tries to calculate the angle. AJ’s work: cos 𝜃 =
!
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Jace looked at his work and exclaimed, “Of course your calculator gives you an error! Your cosine ratio is impossible!” What is Jace talking about? And how could he tell without seeing the triangle or checking on his calculator? Adapted from CPM Core Connections Geometry, Section 5.1.3.