1. 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)

2. Inverse Trigonometric Functions
4.7
Inverse Trigonometric Functions

3. 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.

4. 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

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

6. 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

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

8. 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

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

10. 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

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

12. 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

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

14. 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) = = 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

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

16. 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.

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

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