Further Trig TopicsFurther Trig Topics

Cofunction IdentitiesCofunction Identities

Even and Odd FunctionsEven and Odd Functions

A function is A function is eveneven if if f(x) = f(-x)f(x) = f(-x)

A function is A function is odd odd if if –f(x) = f(-x)–f(x) = f(-x)

Therefore, the cosine function is even Therefore, the cosine function is even and the sine function is odd!and the sine function is odd!

Why is this so you ask? Let me Why is this so you ask? Let me explain!explain!

Consider cos 60Consider cos 60°°

Cos 60° = cos(-60°) , therefore an even Cos 60° = cos(-60°) , therefore an even function function

Consider sin 60°Consider sin 60°

Sin 60° ≠ sin(-60°)Sin 60° ≠ sin(-60°)

Instead, -sin 60° = sin(-60°), therefore oddInstead, -sin 60° = sin(-60°), therefore odd

Cosine GraphCosine Graph

Sine GraphSine Graph

Sine and Cosine GraphsSine and Cosine Graphs

Equivalent FunctionsEquivalent Functions

Trig FunctionTrig Function

f (x) = - sin xf (x) = - sin x

f (x) = cos xf (x) = cos x

f (x) = cos xf (x) = cos x

f (x) = sin xf (x) = sin x

Equivalent Trig FunctionEquivalent Trig Function

f (x) = cos (-x)

f (x) = sin (-x)

f (x) = cos (x - 90° )

f (x) = sin ( x+ 90° )

Equivalent FunctionsEquivalent Functions

Trig FunctionTrig Function Equivalent FunctionEquivalent Function

f(x) = cos(-x)f(x) = cos(-x) f(x) = cos xf(x) = cos x

f(x) = sin(-x)f(x) = sin(-x) f(x) = - sin xf(x) = - sin x

f(x) = cos(x - 90f(x) = cos(x - 90°°)) f(x) = sin xf(x) = sin x

f(x) = sin(x + 90f(x) = sin(x + 90°°)) f(x) = cos xf(x) = cos x

Tangent FunctionTangent Function

Copy and complete the table on page 222Copy and complete the table on page 222Change radian values into degreesChange radian values into degreesDraw vertical lines where tangent is Draw vertical lines where tangent is undefinedundefinedWhen is tangent undefined?When is tangent undefined?Answer: When cosine = 0Answer: When cosine = 0The vertical lines are called “asymptotes”The vertical lines are called “asymptotes”Def – lines that are approached but not Def – lines that are approached but not touched by the curve.touched by the curve.

Tangent GraphTangent Graph

Characteristics of tangent graphCharacteristics of tangent graph

Period?Period?180180° or ° or ππZeroes?Zeroes?0 + 0 + ππkkLocation of asymptotes?Location of asymptotes?ππ/2 + /2 + ππkkWhat is the limit of the function as What is the limit of the function as xx approaches approaches ππ/2?/2?∞∞Is the tangent function odd or even?Is the tangent function odd or even?OddOdd

Transformations of sine:Transformations of sine:

Graph 2(y + 1) = sin 3(x – Graph 2(y + 1) = sin 3(x – ππ/2)/2)

Find the maximum and minimum valuesFind the maximum and minimum values

Give the range and domainGive the range and domain

Find the intervals of increase and Find the intervals of increase and decreasedecrease

Transformations of cosine:Transformations of cosine:

Graph ¼(y - 2) = cos ½(x + Graph ¼(y - 2) = cos ½(x + ππ))

Find the maximum and minimum valuesFind the maximum and minimum values

Give the range and domainGive the range and domain

Find the intervals of increase and Find the intervals of increase and decreasedecrease

Transformations of tangent:Transformations of tangent:

Tips:Tips:1. Draw the sinusoidal axis1. Draw the sinusoidal axis2. Translate the point (0,0) using the sinusoidal 2. Translate the point (0,0) using the sinusoidal axis and the phase shiftaxis and the phase shift3. Use the period to find the location of the 3. Use the period to find the location of the points. Add to obtain other points and draw points. Add to obtain other points and draw asymptotes in the middle.asymptotes in the middle.4. Draw other asymptotes using the period4. Draw other asymptotes using the period5. Draw curves in between asymptotes5. Draw curves in between asymptotes

Transformations of tangent:Transformations of tangent:

Graph y + 1 = tan 3(Graph y + 1 = tan 3(θθ – – ππ))Find the maximum and minimum valuesFind the maximum and minimum valuesState the intervals of increase and State the intervals of increase and decreasedecreaseGive the location of the asymptotesGive the location of the asymptotesFind the range and domainFind the range and domainWhat is the limit as What is the limit as x x approaches approaches the the asymptotes from both sides?asymptotes from both sides?

Another Transformation of Tangent!Another Transformation of Tangent!

Graph y – 3 = tan ¼(Graph y – 3 = tan ¼(θθ + + ππ/2)/2)

Max and min?Max and min?

Intervals of increase and decrease?Intervals of increase and decrease?

Location of asymptotes?Location of asymptotes?

Range and domain?Range and domain?

Limits?Limits?

What is the equation?What is the equation?

Answer?Answer?

y – 3 = tan x y – 3 = tan x ± ± ππ/2/2

Give the range and domainGive the range and domain

{y | y {y | y ЄЄ R} R}

{x | x {x | x ЄЄ R, x ≠ 0 + R, x ≠ 0 + ππkk, , kk ЄЄ I} I}

What is the equation(#2)?What is the equation(#2)?

Answer?Answer?

y + 2 = tan 3(x – y + 2 = tan 3(x – ππ/3)/3)

Give the range and domainGive the range and domain

{y | y {y | y ЄЄ R} R}

{x | x {x | x ЄЄ R, x ≠ R, x ≠ ππ/6 + /6 + ππ/3 /3 kk,, k k ЄЄ I} I}

CotangentCotangent

Cotangent has a similar look to the tangent Cotangent has a similar look to the tangent graphgraphSince it is the reciprocal of tangent, wherever Since it is the reciprocal of tangent, wherever tangent is 0, then cotangent will be undefined tangent is 0, then cotangent will be undefined with asymptoteswith asymptotesThe zeroes of tangent are 0 + The zeroes of tangent are 0 + ππk k so this will be so this will be the asymptotes for cotangentthe asymptotes for cotangentThe direction of the curve will be decreasing The direction of the curve will be decreasing instead of increasinginstead of increasing

Cotangent GraphCotangent Graph

Cotangent FunctionCotangent Function

Tips for graphing:Tips for graphing:1. Since cotangent is the reciprocal of tangent, 1. Since cotangent is the reciprocal of tangent, the asymptotes will be where tangent = 0.the asymptotes will be where tangent = 0.2. Instead of translating (0,0), the asymptote at 2. Instead of translating (0,0), the asymptote at x = 0 will be translated using the phase shiftx = 0 will be translated using the phase shift3. The period will then be added to each side to 3. The period will then be added to each side to find other asymptotes and the graph will be find other asymptotes and the graph will be drawn in between the asymptotesdrawn in between the asymptotes4. The graph goes the opposite direction as 4. The graph goes the opposite direction as tangenttangent

Cotangent GraphsCotangent Graphs

Graph and state the range and domain of:Graph and state the range and domain of:

1. y – 3 = cot 2(x – 1. y – 3 = cot 2(x – ππ/3)/3)

2. y + 1 = cot 1/3(x + 2. y + 1 = cot 1/3(x + ππ/2)/2)

Give the equation in terms of Give the equation in terms of cotangent:cotangent:

Equation of Cotangent GraphEquation of Cotangent Graph

Answer?Answer?

y – 1 = cot 3(x – y – 1 = cot 3(x – ππ/6)/6)

Range and Domain?Range and Domain?

{y {y | y | y ЄЄ R} R}{x | x {x | x ЄЄ R, x ≠ R, x ≠ ππ/6/6k, kk, k ЄЄ I} I}

Cosecant and Secant GraphsCosecant and Secant Graphs

Copy and complete the chart on page 233Copy and complete the chart on page 233

Graph the sine valuesGraph the sine values

Graph the cosecant valuesGraph the cosecant values

Where will the asymptotes be located?Where will the asymptotes be located?

Answer : where sine = 0Answer : where sine = 0

Sine and cosecant graphsSine and cosecant graphs

Characteristics of CosecantCharacteristics of Cosecant

Period?Period?

22ππ

Maximum and Minimum?Maximum and Minimum?

∞ ∞ and -∞and -∞

Local maxima and minima?Local maxima and minima?

-1 and 1-1 and 1

Cosine and Secant GraphsCosine and Secant Graphs

Transformations of Sec and CscTransformations of Sec and Csc

For each of the following, graph and find For each of the following, graph and find thethe1. Range and Domain1. Range and Domain2. Local maxima and minima2. Local maxima and minima

y + 1 = csc ½(x – y + 1 = csc ½(x – ππ/2)/2)

¼(y – 2) = sec 2(x – ¼(y – 2) = sec 2(x – ππ))

Answer!Answer!

Answer!Answer!

Compound Angle IdentitiesCompound Angle Identities

sin(a + b) = sin a cos b + cos a sin bsin(a + b) = sin a cos b + cos a sin b

cos(a + b) = cos a cos b – sin a sin bcos(a + b) = cos a cos b – sin a sin b

Example 1:Example 1:

Evaluate sin 75Evaluate sin 75° in exact form:° in exact form:

sin(45° + 30°) = sin 45 cos 30 + cos 45 sin 30sin(45° + 30°) = sin 45 cos 30 + cos 45 sin 30

= = ((21