Graphs of Tangent, Cotangent, Secant, and Cosecant 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

Quiz: 4-4 1. Vertical stretch/shrink 2. Horizontal stretch/shrink 3. Horizontal translation (phase shift) 4. Vertical translation

What you’ll learn about The Tangent Function The Cotangent Function The Secant Function The Cosecant Function … and why This will give us functions for the remaining trigonometric ratios.

UNIT CIRCLE 30º A C B 1 60º 3 2 C r = 1 60º 2 y 30º A x B

UNIT CIRCLE Vertical asymptote 1 -90º -60º -30º 0º 30º 60º 90º -1

UNIT CIRCLE Pattern repeats 1 -90º 0º 90º -1

Your turn: Your turn: Domain = ? Range = ? Continuous? Symmetry? bounded? Vertical/horizontal asymptotes? End behavior?

f(x) = sin x Remember this? radians radians The y-coordinate of the graph is the distance above/below the x-axis on the unit circle. The x-coordinate of the graph is the distance around the unit circle. Same thing for cos x.

How do we construct the Tangent Function’s graph? Sin x y = Sin x: the vertical distance above/below the x-axis (on the unit circle). This distance is graphed as the y-coordinate on this graph. x = cos x: the horizontal distance to the right/left of the y-axis (on the unit circle). Cos x This distance is also graphed as the y-coordinate on this graph.

Construct the Tangent Function using the graph of ‘y’ and ‘x’ y = Sin x: (on unit circle) y = Sin x x = cos x: (on unit circle) Using the graphs of the left: x = Cos x

The Tangent and Cotangent Functions Tan x Cot x

Asymptotes and Zeros of the Tangent Function X-intercepts: sin x = 0 Asymptotes: cos x = 0 Rules for Tangent: y = tan (bx – c) + d a = none period = A full period of tan occurs between 2 consecutive asymptotes. To find 2 consecutive asymptotes: bx – c = and bx – c = The graph starts low, ends high, ½ way thru period is where it crosses the x-axis.

The Tangent Function X-intercepts: sin x = 0 Asymptotes: cos x = 0 Amplitude: doesn’t have one (look at the graph) A full period of tan(x) occurs between 2 asymptotes which occur at odd multiples of Period: Shrink factor causes asymptotes to be odd multiplies of:

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

The Sine and Cosecant Functions Sine (x) Cosecant (x)

The Cosecant Function: y = 2 csc 2x To graph graphing shifts, rewrite the equation as it’s reciprocal, then find zero’s, max/mins (amp), and period. 1. Vertical stretch: factor of 2 2. Vertical asymptotes: even multiples of 3. Horizontal shrink factor of ½, changes period to sin(x) csc(x) 2csc(2x)

The Cosine and Secant Functions Cos x Sec x

Solve Trig function Algebraically Find: ‘x’ such that and sec(x) = -2 Converting to unit circle: r = 1 x = -½ That looks familiar! 3 30º 1 60º ½ 2 -½ 60º 1 240º x = 240º

Solving a Trig equation graphically Find the smallest number ‘x’ such that: 1. Graph and 2. Find the point of intersection

Basic Trigonometry Functions

HOMEWORK Section 4-5