Chapter 14 Trigonometric Graphs, Identities, and Equations Math III Accelerated Chapter 14 Trigonometric Graphs, Identities, and Equations

Warm Up 14.1 Find the sine, cosine, and tangent for each.

14.1 Graph Sine, Cosine, and Tangent Functions Objective: Graph sine, cosine, and tangent functions.

Periodic Function A function is periodic if it repeats at regular intervals. A cycle is the shortest repeating portion. The period is the horizontal length of one cycle. The amplitude is one-half the difference of the maximum, M and the minimum, m.

Examples of Periodic Functions

Sine Create a table of values for y = sin x & graph. x 0° 30° 45° 60° 90° 135° 180° 270° 360° sin x  

Graph of y = sin x Domain: Range: Period: Amplitude: Max: Min: Zeros:

Sine Function The function y = sin x is the parent function. A simplified version of the general equation for sine is where |a| = amplitude b = number of cycles in one period That is, period =

Graph y = sin x By Hand Use five key points in one period: Zero, Max, Zero, Min, Zero

Example 1 Graph y = 2 sin x  

Example 2 Graph y = sin 4x  

Cosine Create a table of values for y = cos x & graph. x π 2π cos x  

Graph of y = cos x Domain: Range: Period: Amplitude: Max: Min: Zeros:

Cosine Function The function y = cos x is the parent function. A simplified version of the general equation for cosine is where |a| = amplitude b = number of cycles in one period That is, period =

Graph y = cos x By Hand Use five key points in one period: Max, Zero, Min, Zero, Max

Example 3 Graph  

Example 4 Graph y = –3 cos x  

Example 5 (NTG Ex. 2) Write a sine function with an amplitude of 3 and a frequency of 1000.

Tangent Create a table of values for y = tan x & graph. x –1.57 –1.5 1.5 1.57 tan x  

Graph of y = tan x Domain: Range: Period: Zeros: VAs:

Tangent Function The function y = tan x is the parent function. The simplified version of the general equation for tangent is where b = number of cycles in one period That is, period =

Graph y = tan x By Hand Find the VAs Solve for x: and . Period is distance between VAs. Find the zero Midway between the VAs

Example 6 (NTG Ex. 3) Graph one period of the function y = 2 tan x.  

Example 7 (NTG CP 5) Graph one period of the function y = tan 4x.  

Example 8 (NTG CP 6) Graph one period of the function y = tan πx.  

Homework 14.1 Practice 14.1