Trigonometric Functions

Examples Find Sine, Cosine and Tangent of θ Sine = 12/15 = .8 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 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

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

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

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

(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)

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) = 14 7 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