1. Trigonometric Functions

2. Examples Find Sine, Cosine and Tangent of θ Sine = 12/15 = .8
Tangent = 12/9 = 1.33 (or 4/3)

3. Reciprocals of Sin/Cos/Tan
Reciprocal of Sine is Cosecant = 1/Sin
Hypotenuse over Opposite : csc
Reciprocal of Cosine is Secant = 1/Cos
Hypotenuse over Adjacent : sec
Reciprocal of Tangent is Cotangent = 1/Tan
Adjacent over Opposite : cot

4. Examples Find Csc, Sec and Cot of Θ Csc = 15/12 = 1.25
Sec = 15/9 = 1.66 (or 5/3)
Cot = 9/12 = .75

5. Angles of Rotation Standard Position Initial Side Terminal Side
Vertex is origin
One ray is positive x axis
Initial Side
Terminal Side
Angle of Rotation
Maintain initial side and rotate to terminal side

6. Reference Angle Positive acute angle of the triangle
Quadrant of Reference angle determines sign of functions

7. Sine, Cosine, Tangent For a RIGHT TRIANGLE SOH CAH TOA
Sine – Opposite over Hypotenuse : sin
Cosine – Adjacent over Hypotenuse : cos
Tangent – Opposite over Adjacent : tan
SOH
CAH
TOA

8. Trig to Circles
If vertex is (0,0) - trig uses x and y coordinates of point
Radius (r) is √(x2+y2) : (Sqrt of x2+y2)
Sine is y/r, Cosine is x/r, and Tangent is y/x

9. (3, 4) (-3, 4) (-3, -4) (3, -4) Examples
Use the following coordinates to determine the trigonometric functions (sin, cos, tan):
(3, 4)
(-3, 4)
(-3, -4)
(3, -4)

10. Signs in Quadrants
The location of the reference angle determines the sign of the functions

11. Inverse Trig Functions
Going from value to angle measure
On calculator – sin-1(a) or cos-1(a) or tan-1(a)
Get there by 2nd SIN/COS/TAN then enter the value in the parentheses
Value for sin/cos must be -1≤a≤1
Example:
Find m<θ
: sinθ = 7/14
: sinθ = .5
: sin-1(.5) = 14 7 θ

12. Restrictions on Inverse Functions
Domains & Ranges are restricted as follows:

13. Special Right Triangles
30/60/90
45/45/90

14. Unit Circle Circle with a radius of 1
Relation of radians, degrees and the sine and cosine of the related angles
Coordinates of point on circle are (cosθ, sinθ)
Cosine is the x coordinate
Sine is the y coordinate

15. Unit Circle

16. Radians and Degrees Radian – Angle measure based on arc length
Circumference of circle = 2πr
Complete revolution of circle = 360o
Relationship of radians to degrees is 2π = 3600

17. Graphing Sin/Cos Functions
Periodic – repeats exactly at a given interval
Intervals are called cycles
Length of the cycle is the period
Sin & Cos are Periodic
Values are the y & x
values on unit circle
Period is 2π -
1 complete rotation

18. Transformations Period (cycle length) and Amplitude (height)
y = a sin bx or y = a cos bx
a is the amplitude – absolute value (positive)
2π/b is the period
Phase Shift - function left/right or up/down
h (left/right) and k (up/down) values in function

19. Trigonometric Identities
Use to compare and simplify trigonometric functions
Based on following table and algebraic solving

20. Trig Identity Examples
: sinθcotθ = cosθ :
: secθ – tanθ sinθ
Using calculator :
Enter into Y1 & Y2
Compare Graphs