Chapter 5 Sections 2.3 – 2.4 – 2.5

Y = SIN X X is your Angle (θ) Y is your Height

Table

y = Sin(x) 0 < x < 2π

y = sin x,

Definition

Y = Sin x Sin(-x) = - Sin(x) ODD FUNCTION: Symmetric About Origin Like f(x) = x3 Sin(-x) = - Sin(x) Example: Sin(- 𝜋 2 )=−1=−𝑆𝑖𝑛( 𝜋 2 )

Y = Sin x Sin(-2) = Sin(0) = Sin(2) = Sin(2n), for any Integer n Period = 2𝜋 Sin(-2) = Sin(0) = Sin(2) = Sin(2n), for any Integer n Sin(θ) = Sin(θ + 2n) , for any Integer n FUNDAMENTAL PERIOD: First Complete Cycle (2𝜋) Amplitute: Max Absolute Value of Height = 1

Graph of y= csc x = 1 𝑆𝑖𝑛 𝑥 Domain: ALL REALS, Except Integer Multiples of  Range: (-∞, -1] U [1, ∞) NO AMPLITUTE Period = 2

Y = COS X X is your Angle (θ) Y is your Height

Table

y = cos x, 0 < x < 2π

Figure: y = cos x,

Y = Cos x Cos(-x) = Cos(x) EVEN FUNCTION: Symmetric About Y-Axis Like f(x) = x2 Cos(-x) = Cos(x) Example: Cos(- 𝜋 2 )=0=Cos( 𝜋 2 )

Y = Cos x Cos(-2) = Cos(0) = Cos(2) = Cos(2n), for any Integer n Period = 2𝜋 Cos(-2) = Cos(0) = Cos(2) = Cos(2n), for any Integer n Cos(θ) = Cos(θ + 2n) , for any Integer n FUNDAMENTAL PERIOD: First Complete Cycle (2𝝅) Amplitute: Max Absolute Value of Height = 1

Graph of y= sec x = 1 𝐶𝑜𝑠 𝑥 Domain: ALL REALS, Except Odd Integer Multiples of 𝜋 2 Range: (-∞, -1] U [1, ∞) NO AMPLITUTE Period = 2

Y = Tan X = 𝑆𝑖𝑛 𝑋 𝐶𝑜𝑠 𝑋 X is your Angle (θ) Y is your Height

Graph of

Table

Figure: y = tan x

Y = Tan x Tan(-x) = - Tan(x) ODD FUNCTION: Symmetric About Origin Like f(x) = x3 Tan(-x) = - Tan(x) Example: Tan(- 𝜋 4 )=−1=−Tan( 𝜋 4 )

Y = Tan x Tan(-) = Tan(0) = Tan() = Tan(n), for any Integer n Period = 𝜋 Tan(-) = Tan(0) = Tan() = Tan(n), for any Integer n Tan(θ) = Tan(θ + n) , for any Integer n FUNDAMENTAL PERIOD: First Complete Cycle (𝜋) Amplitute: NONE Domain: All Real Numbers, Except Odd Multiples of 𝜋 2 Range: (-∞, ∞)

Table: y = cot x

Figure: Graph of y = cot x

Y = Cot x = 1 𝑇𝑎𝑛 𝑋 Cot(-x) = - Cot(x) ODD FUNCTION: Symmetric About Origin Like f(x) = x3 Cot(-x) = - Cot(x) Example: Cot(- 𝜋 4 )=−1=−Cot( 𝜋 4 )

Y = Cot x Period = 𝜋 Cot(-3/2) = Cot(-/2) = Tan(/2) = Tan(n/2), for any Integer n Cot(θ) = Cot(θ + n) , for any Integer n FUNDAMENTAL PERIOD: First Complete Cycle (𝜋) Amplitute: NONE Domain: All Real Numbers, Except Odd Multiples of  Range: (-∞, ∞)

Table

Theorem

Graph Functions of the Form y=A sin(wx) Using Transformations

Solution

Figure: y = - sin(2x)

Graph Functions of the Form y=A cos(wx) Using Transformations

Example

Solution

Figure

Example

Solution

Find an Equation for a Sinusoidal Graph

Figure

Example Figure 94

Solution

Graph Functions of the Form y=A tan(wx)+B and y=A cot(wx)+B

Example

Solution Figure 97

Example

Solution

Solution continued

Graph Functions of the Form y=A csc(wx)+B and y=A sec(wx)+B

Example

Solution Figure 102

Example

Solution

Figure

Determine the Signs of the Trigonometric Functions in a Quadrant

Figure

Table

Figure

Example

Solution

Use Even-Odd Properties to Find the Exact Values of the Trigonometric Functions

Figure

Example

Solution

Find the Values of the Trigonometric Functions Using Fundamental Identities

Example

Solution

Example

Solution

Find the Exact Values of the Trigonometric Functions of an Angle Given One of the Functions and the Quadrant of the Angle

Example

Solution Option 1 Using a Circle

Solution Option 1 Using a Circle continued

Solution Option 2 Using Identities

Solution Option 2 Using Identities continued

Example

Solution Option 1 Using a Circle

Figure

Solution Option 2 Using Identities

Solution Option 2 Using Identities continued