Inverse Trigonometric Functions 4.7 By: Ben O’Shasky, Luke Gause, and Tyler Gilbert

Introduction to Inverse Trigonometric Functions The Inverse Trigonometric Functions are the inverse functions of the Trigonometric Functions. They are viewed as: sin−1, cos−1, tan−1 . You can evaluate them on your calculator or evaluate them without your calculator which we will be showing through our examples later on. You use the Inverse functions when you have the trigonometric ratio and you need to find the angle that has that trigonometric ratio.

Unit Circle (0, 1) (1, 0) (-1, 0) (0, -1) 60º 45º Red= 30º angles Note: Cosine is negative in the II and III quadrant. Sine is negative in the III and IV quadrant. Tangent is negative in the II and IV quadrant. (0, 1) 60º 45º Red= 30º angles Blue= 45º angles Green= 60º angles Purple= 90º angles 30º (1, 0) (-1, 0) X= Cosine Y= Sine Y/X= Tangent (0, -1) http://www.youtube.com/watch?v=WTz6PQIFNek

Unit Triangles 45, 45, 90 Triangle 30, 60, 90 Triangle 45º √2 60º √3 1 30º 45º 2 1 ***Use the 45, 45, 90 triangle when you are dealing with an angle that is a 45º angle. Use the 30, 60, 90 triangle when you are dealing with a 30º or a 60º angle. Match the trigonometric ratio you are given (depending on if you are using sine, cosine, or tangent) to the angle (X) that you are trying to find.

Evaluating Angles Using Inverse Y=sin-1 X X=sin Y Example: sin-1 (1/√ 3) = X Step 1: Figure out which triangle to use The fraction has numbers from the 30, 60, 90 triangle. Since sine is Opposite side over Hypotenuse, the angle we are looking for must match this fraction. Step 2: The angle that has an opposite of 1 and has the hypotenuse of √3 is the 30º angle. Therefore the X= 30º The process is the same for cosine and tangent. Cosine is adjacent/hypotinuse Tangent is opposite/adjacent √ 3 60º 1 30º 2

Evaluating Inverse Using Calculator Steps on a Calculator 2nd Sin, cos, or tan (depending on what inverse you are using) Type in the trigonometric ratio. Make sure you have parentheses around ratio when needed. Example: cos(X) = 1/2 cos-1 (1/2) = X Follow the calculator steps Cos-1 (1/2) Press “ENTER” to get the angle. In this example…. 60 Unknown Angle Trigonometric Ratio

Calculating a viewing angle In order to calculate a viewing angle you need to know two side lengths of a triangle Ex.) 10 6 x Arcsin(10/6) = x X = 36.870

tan-1(1) cos-1(1/2) sin-1(1/√2) sin-1(1/2) tan-1(√3) Find the Exact Value Without using a calculator, than check your answers using a calculator. tan-1(1) cos-1(1/2) sin-1(1/√2) sin-1(1/2) tan-1(√3)

Flash Card Answers π /4 or 45º π /3 or 60º π /6 or 30º

Multiple Choice Questions Find the exact value without a calculator for the ones without decimals and a calculator for the ones with decimals… Arcsin(√3/2) Arc sin(-1/√2) Arc cos(1/2) Arc sin( ½) Arctan (1/4) Arc cos( 3/5) Arcsin(2/3) Arctan(1/6) Arccos(5/6) Arctan(383/500)

Multiple Choice A)60 degrees B) 45 degrees C) 30 degrees D) 90 degrees

Multiple Choice Continued A) 50 degrees b) 75 degrees c) 30 degrees d) 105 degrees A) 19.048 degrees B)14.036 degrees C) 29.476 degrees D) 68.934 degrees A) 53.130 degrees B)72.541degrees C)92.354 degrees D)13.458 degrees

Multiple Choice Continued 7. A) 10.341π B) .232π C).934π D)1.567π A) 9.462 B) 23.458 C)2.537 D)49.321 A) 51.324 B)11.254 C)89.352 D)33.557 A)50 B)75.284 C)60.302 D)37.452

Answers A C B D

References http.//www.youtube.com/watch?v=WTz6PQIFNek http://www.coolmath.com/precalculus-review-calculus- intro/precalculus-trigonometry/28-the-unit-circle-01.htm http://seyyalsarac.com/2/trig-comics Precalculus Graphical, Numerical, Algebraical (book)