Trigonometric Functions

Examples

• Find Sine, Cosine and Tangent of θ

• Sine = 12/15 = .8• Cosine = 9/15 = .6 • Tangent = 12/9 = 1.33 (or 4/3)

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

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

Angles of Rotation

• Standard Position – Vertex is origin – One ray is positive x axis

• Initial Side• Terminal Side

• Angle of Rotation – Maintain initial side and rotate to terminal side

Reference Angle

• Positive acute angle of the triangle• Quadrant of Reference angle determines sign

of functions

Sine, Cosine, Tangent• For a RIGHT TRIANGLE– Sine – Opposite over Hypotenuse : sin– Cosine – Adjacent over Hypotenuse : cos– Tangent – Opposite over Adjacent : tan

• SOH• CAH• TOA

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

Examples

• Use the following coordinates to determine the trigonometric functions (sin, cos, tan):

1. (3, 4)2. (-3, 4)3. (-3, -4)4. (3, -4)

Signs in Quadrants

• The location of the reference angle determines the sign of the functions

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) =

7 14

θ

Restrictions on Inverse Functions

• Domains & Ranges are restricted as follows:

Special Right Triangles

• 30/60/90• 45/45/90

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

Unit Circle

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

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

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

Trigonometric Identities

• Use to compare and simplify trigonometric functions

• Based on following table and algebraic solving

Trig Identity Examples

• : sinθcotθ = cosθ

• :

• : secθ – tanθ sinθ

• Using calculator : – Enter into Y1 & Y2

– Compare Graphs