Quiz 13-3 1. Draw the standard position angle for 210º 2. What is the reference angle for 210º ? 3. What triangle do we need to solve? (label angle and the length of each of the sides) 4. Is cos 210º (+) or (-) ? 5. Cos 210º = ?

Quiz 13-3 -x =½ 4. Is cos 210º (+) or (-) ? 1. Draw the standard position angle for 210º 2. What is the reference angle for 210º ? 3. What triangle do we need to solve? (label angle and the length of each of the sides) 90º 5. Cos 210º = ? -x 180º 0º x 30º 30º 210º y =½ 60º 1 270º

Inverse Trigonometric Functions 13-4 Inverse Trigonometric Functions

What you’ll learn about (13-4) 1. What is an Inverse trigonometric function? How to use Inverse Trigonometric Functions to solve for the unknown angles in a triangle (using a calculator). How to solve trigonometric equations using inverse trig. functions. 4. How to evaluate inverse trigonometric functions without using a calculator.

Compositions of Functions (ch-6 review) In a composition of functions, instead of having a number as an input value, you have another function as an input value. 1. Your turn:

Compositions of Inverse Functions The composition of inverse functions “cancels out” each of the functions leaving: input value = output value

Your turn: 2. f(g(x) = ? 3. f(g(x) = ?

Review of Section 13-1 B Sin A = ? 17 8 Cos B = ? A C 15 Tan A = ?

Given: the angle and a side Find: unknown sides Which function would you use to find ‘y’ ? 8 y ‘y’ = ? 25º x Which function would you use to find ‘x’ ? 4. y = ? Your turn: 5. x = ? ‘x’ = ?

Given: the length of the sides Find: the angles 3 4 5 How do we find the angle? If we could compose with its inverse function, we could then have all by itself  we would know the angle. What is the inverse function for sine?

Inverse function for sine, cosine, tangent When you compose a function with its inverse function: output = input. Look at the “sin” button on your calculator. Notice the function right above the “sin” function. 3 4 5

Another example Use the sine ratio to find: 13 5 12 25 Your turn: 7 6. 24

Another way to think about it: Sine of an angle = a ratio Inverse sine of an ratio = an angle 25 7 24

Your turn: 26 7. Use: 10 24 to find the measure of the angle.

REMEMBER!!! Sine (angle) = ratio Inverse sine (ratio) = angle

Solving Trigonometric Equations using Inverse Functions. Solve: REMEMBER!!! Inverse sine (ratio) = angle Which triangle applies (45-45-90 or 30-60-90) ? 60º 1 45º ½ 1 30º 45º Reference angle = 45º

For what angles will the sine ratio be negative? 90 135º 45º 1 45º 180 45º 45º 45º 225º 315º 225º 315º 270

Solving Trigonometric Equations using Inverse Functions. Solve: Inverse cosine (ratio) = angle Which triangle applies (45-45-90 or 30-60-90) ? 60º 1 45º ½ 1 30º 45º Reference angle = 30º

will the cosine ratio be positive? 90 For what angles will the cosine ratio be positive? 90 30º 150º 1 180 30º 30º 30º 30º 30º 300º 210º 300º 270

Your Turn: Solve: (step by step) 8. What triangle applies? What is the reference angle? Draw the unit circle with the 4 possible triangles in position. What are the angles where the tangent ratio will be negative?

Word Problem: An airplane is flying at 35,000 feet. When it is 100 miles away from the Salt Lake City airport it begins descending. Assuming the airplane descends at the same angle for the whole distance, what is the angle of descent? 1. Draw the picture: 35,000 ft 100 miles 2. Do you need to convert any units? Either 35,000 ft to miles or 100 miles to feet.

Word Problem: 1. Draw the picture: 35,000 ft 100 miles 2. Convert units 6.63 miles 100 miles 3. Set up equation. 4. Are you in angle mode? 5. Solve