1. Cofunction Identities
Further Trig Topics
Cofunction Identities

2. Even and Odd Functions A function is even if f(x) = f(-x)
A function is odd if –f(x) = f(-x)
Therefore, the cosine function is even and the sine function is odd!

3. Why is this so you ask? Let me explain!
Consider cos 60°
Cos 60° = cos(-60°) , therefore an even function
Consider sin 60°
Sin 60° ≠ sin(-60°)
Instead, -sin 60° = sin(-60°), therefore odd

4. Cosine Graph

5. Sine Graph

6. Sine and Cosine Graphs

8. Equivalent Trig Function
Equivalent Functions
f (x) = cos (-x)
f (x) = sin (-x)
f (x) = cos (x - 90° )
f (x) = sin ( x+ 90° )
Trig Function
Equivalent Trig Function
f (x) = cos x
f (x) = - sin x
f (x) = sin x
f (x) = cos x

9. Equivalent Functions Trig Function Equivalent Function f(x) = cos(-x)
f(x) = sin(-x)
f(x) = - sin x
f(x) = cos(x - 90°)
f(x) = sin x
f(x) = sin(x + 90°)

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

11. Tangent Graph

12. Characteristics of tangent graph
Period?
180° or π
Zeroes?
0 + πk
Location of asymptotes?
π/2 + πk
What is the limit of the function as x approaches π/2? ∞
Is the tangent function odd or even?
Odd

13. Transformations of sine:
Graph 2(y + 1) = sin 3(x – π/2)
Find the maximum and minimum values
Give the range and domain
Find the intervals of increase and decrease

14. Transformations of cosine:
Graph ¼(y - 2) = cos ½(x + π)
Find the maximum and minimum values
Give the range and domain
Find the intervals of increase and decrease

15. Transformations of tangent:
Tips:
1. Draw the sinusoidal axis
2. Translate the point (0,0) using the sinusoidal axis and the phase shift
3. Use the period to find the location of the points. Add to obtain other points and draw asymptotes in the middle.
4. Draw other asymptotes using the period
5. Draw curves in between asymptotes

16. Transformations of tangent:
Graph y + 1 = tan 3(θ – π)
Find the maximum and minimum values
State the intervals of increase and decrease
Give the location of the asymptotes
Find the range and domain
What is the limit as x approaches the asymptotes from both sides?

17. Another Transformation of Tangent!
Graph y – 3 = tan ¼(θ + π/2)
Max and min?
Intervals of increase and decrease?
Location of asymptotes?
Range and domain?
Limits?

18. What is the equation?

19. Answer? y – 3 = tan x ± π/2 Give the range and domain {y | y Є R}
{x | x Є R, x ≠ 0 + πk, k Є I}

20. What is the equation(#2)?

21. Answer? y + 2 = tan 3(x – π/3) Give the range and domain {y | y Є R}
{x | x Є R, x ≠ π/6 + π/3 k, k Є I}

22. Cotangent Cotangent has a similar look to the tangent graph
Since it is the reciprocal of tangent, wherever tangent is 0, then cotangent will be undefined with asymptotes
The zeroes of tangent are 0 + πk so this will be the asymptotes for cotangent
The direction of the curve will be decreasing instead of increasing

23. Cotangent Graph

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

25. Cotangent Graphs Graph and state the range and domain of:
1. y – 3 = cot 2(x – π/3)
2. y + 1 = cot 1/3(x + π/2)

26. Give the equation in terms of cotangent:

27. Equation of Cotangent Graph
Answer?
y – 1 = cot 3(x – π/6)
Range and Domain?
{y | y Є R}
{x | x Є R, x ≠ π/6k, k Є I}

28. Cosecant and Secant Graphs
Copy and complete the chart on page 233
Graph the sine values
Graph the cosecant values
Where will the asymptotes be located?
Answer : where sine = 0

29. Sine and cosecant graphs

30. Characteristics of Cosecant
Period? 2π
Maximum and Minimum?
∞ and -∞
Local maxima and minima?
-1 and 1

31. Cosine and Secant Graphs

32. Transformations of Sec and Csc
For each of the following, graph and find the
1. Range and Domain
2. Local maxima and minima
y + 1 = csc ½(x – π/2)
¼(y – 2) = sec 2(x – π)

33. Answer!

34. Answer!

35. Compound Angle Identities
sin(a + b) = sin a cos b + cos a sin b
cos(a + b) = cos a cos b – sin a sin b
Example 1:
Evaluate sin 75° in exact form:
sin(45° + 30°) = sin 45 cos 30 + cos 45 sin 30
= (