Quiz 4-5 Describe how tan(x) is transformed to graph: -tan(2x) 1. Amplitude: . Amplitude: n/a, -1 coefficient reflection across x-axis. 2. Period: 3. Vertical Asymptotes: odd multiples of Tan(x) -tan(2x)

Inverse Trigonometric Functions 4.7 Inverse Trigonometric Functions

What you’ll learn about Inverse Sine Function Inverse Cosine and Tangent Functions Composing Trigonometric and Inverse Trigonometric Functions Applications of Inverse Trigonometric Functions … and why Inverse trig functions can be used to solve trigonometric equations.

Inverse Sine Function Every function has an inverse. Sin(x): the ratio of the side opposite the angle to the hypotenuse of a right triangle containing the opp. side. the angle that matches a given ratio of side opposite the angle to the hypotenuse. If we consider the interval , then sin(x) and its inverse are one-to-one functions (every output has exactly one input and, due to being a function, every input has exactly one output. Notice the different domains/ranges

Inverse Sine Function (Arcsine Function) 3 30º 1 60º ½ 2

Examples: Find the exact value (use unit circle) of (in degrees): 45º Which is it? Remember, we use sine ‘x’ on the interval: 315º 225º Answer: 315

Your turn: 1. Find the exact value of: without use a calculator in degrees. 2. What is the answer in radians?

Inverse Cosine Function cos(x): the ratio of the side adjacent the angle to the hypotenuse of a right triangle containing the adj. side. the angle that matches a given ratio of side opposite the angle to the hypotenuse. If we consider the interval , then cos(x) and its inverse are one-to-one functions (every output has exactly one input and, due to being a function, every input has exactly one output. 2. M = 55, r = 7, y = ? Notice the different domains/ranges

Inverse Cosine (Arccosine Function) 3 30º 1 60º ½ 2

Inverse Tangent Function (Arctangent Function) tan(x): the ratio of the side opposite to the side adjacent to the angle of a right triangle containing the two sides. the angle that matches a given ratio of opp. side to adj. side in a right triangle. If we consider the interval , then tan(x) and its inverse are one-to-one functions (every output has exactly one input and, due to being a function, every input has exactly one output. Notice the different domains/ranges

Inverse Tangent Function (Arctangent Function) 3 30º 1 60º ½ 2

Examples: 2. Find the exact value (use unit circle) of (in degrees): 30º Which is it? 30º 30º Remember, we use tan ‘x’ on the interval: so we use the same interval for cot(x). 210º Answer: 30

Your turn: 3. Find the exact value of: without use a calculator in degrees. 4. What is the answer in radians?

How do we use the “arcsine function” to solve problems? If you have the angle ‘x’: sin(x) gives you the ratio ‘y/r’. r y y/r = sin(37) = 0.6018 = 3/5 37º If you have the ratio ‘y/r’: arcsin(y/r) gives you the angle ‘x’. 5 3 xº = arcsin(3/5) = 37º xº

Your turn: r y x 5. m = 20, y/r = ? 6. m = 55, r = 7, y = ? 7. r/y = 0.7952, m = ? 8. r = 17, y = 7, m = ?

Composition of Trigonometric functions: Think of a function and its inverse function of “undoing” each other. Whenever you compose a function and its inverse, they “cancel” each other, leaving just the variable.

Trig functions composed with other trig functions that are not inverses. 1 y mº x By substitution:

HOMEWORK Section 4-7 (page 421) (evens) 2-8, 14-20, 24-30, 34-44, 48, 50 (20 problems)