1. 8.4 Relationships Among the Functions
Objective
To simplify trigonometric expressions and to prove trigonometric identities.
To use the fundamental identities to find the values of other trigonometric functions from the value of a given trigonometric function.

2. RECIPROCAL IDENTITIES
QUOTIENT IDENTITIES
PYTHAGOREAN IDENTITIES
Note It will be necessary to recognize alternative forms of the identities above, such as sin²  = 1 – cos²  and cos²  = 1 – sin² .

3. NEGATIVE IDENTITIES
All of the identities we learned are found in the back page of your book under the heading Trigonometric Identities and then Fundamental Identities.
You'll need to have these memorized or be able to derive them for this course.

4. Relationships Among the Functions
Pythagorean Identities
Note It will be necessary to recognize alternative forms of the identities
above, such as sin²  = 1 – cos²  and cos²  = 1 – sin² .

5. Cofunction Relationships
COFUNCTION IDENTITIES
Cofunction Relationships
COFUNCTION IDENTITIES

6. Example 1: Simplify secx – sinx tanx
Each of the trigonometric relationships given is true for all values of the variable for which each side of the equation is defined.
Such relationships are called trigonometric identities.
Example 1: Simplify secx – sinx tanx
[Solution]

7. Example 2: Write tan  + cot  in terms of sin  and cos .
[Solution]

8. [Proof]

9. An Identity is NOT a Conditional Equation
Conditional equations are true only for some values of the variable.
You learned to solve conditional equations in Algebra by “balancing steps,” such as adding the same thing to both sides, or taking the square root of both sides.
We are not “solving” identities so we must approach identities differently.

10. Example 4: Prove
[Proof] Manipulate right to look like left. Expand the binomial and express in terms of sin & cos.

11. We Verify (or Prove) Identities by Doing the Following:
Learn the fundamental identities.
Try to rewrite the more complicated side of the equation so that it is identical to the simpler side.
It is often helpful to express all functions in terms of sine and cosine and then simplify the result.
Usually, any factoring or indicated algebraic operations should be performed. For example,
As you select substitutions, keep in mind the side you are not changing, because it represents your goal.
If an expression contains 1 + sin x, multiplying both numerator and denominator by 1 – sin x would give 1 – sin² x, which could be replaced with cos² x.

12. Suggestions Start with the more complicated side
Try substituting basic identities (changing all functions to be in terms of sine and cosine may make things easier)
Try algebra: factor, multiply, add, simplify, split up fractions
If you’re really stuck make sure to:
Change everything on both sides to sine and cosine.
Work with only one side at a time!

13. cot x + 1 = csc x(cos x + sin x)
Verifying an Identity (Working with One Side)
Example 5: Verify that the following equation is an identity.
cot x + 1 = csc x(cos x + sin x)
Analytic Solution Since the side on the right is more
complicated, we work with it.
Original identity
Distributive property

14. Verifying an Identity (Working with One Side)
Example 6: Verify that the following equation is an identity.
[Proof]

15. Verifying an Identity (Working with Both Sides)
Example 7: Verify that the following equation is an identity.
[Proof]

16. Verifying an Identity (Working with Both Sides)
Now work on the right side of the original equation.
We have shown that

17. How to get proficient at verifying identities:
Once you have proved an identity go back to it, redo the verification or proof without looking at how you did it before, this will make you more comfortable with the steps you should take.
Redo the examples done in class using the same approach, this will help you build confidence in your instincts!

18. Don’t Get Discouraged! Every identity is different
Keep trying different approaches
The more you practice, the easier it will be to figure out efficient techniques
If a solution eludes you at first, sleep on it! Try again the next day. Don’t give up!
You will succeed!

19. If the angle  is acute (less than 90o) and you have the value of one of the six trigonometry functions, you can find the other five.
Reciprocal of sine so "flip" sine over
Sine is the ratio of which sides of a right triangle?
When you know 2 sides of a right triangle you can always find the 3rd with the Pythagorean theorem.
Now find the other trig functions  3 a
"flipped" cos 1
"flipped" tan
Draw a right triangle and label  and the sides you know.

20. We'd still get csc by taking reciprocal of sin
There is another method for finding the other 5 trig functions of an acute angle when you know one function. This method is to use fundamental identities.
We'd still get csc by taking reciprocal of sin
Now use the trig identity
Sub in the value of sine that you know
Solve this for cos 
This matches the answer we got with the other method
square root both sides
We won't worry about  because angle is acute.
You can easily find sec by taking reciprocal of cos.

21. Let's list what we have so far:
We need to get tangent using fundamental identities.
Simplify by inverting and multiplying
Finally you can find cot by taking the reciprocal of this answer.

22. Example 8: If and  is in quadrant II, find each function value.
[Solution]
a) sec
To find the value of this function, look for an identity that relates tangent and secant.
When  is in quadrant II, cos, sec, tan, cot, and csc are all negative.
Tip: Use Pythagorean Identities. 22

23. Example 8: If and  is in quadrant II, find each function value. (Cont
[Solution]
b) sin
Tip: Use Quotient Identities.
c) cot
Tip: Use Reciprocal
d) csc
Tip: Use Reciprocal 23

24. Example 8: If and  is in quadrant II, find each function value. (Cont
[Solution]
e) cos
Tip: Use Reciprocal. 24

25. Example 9: If and  is in quadrant VI, find each function value.
[Solution]
a) cos
Tip: Use Pythagorean Identities.
b) sec
Tip: Use Reciprocal Identities.
c) tan
Tip: Use Quotient Identities.
When  is in quadrant VI, csc, tan, and cot, are all negative, only cos and sec are positive.

26. Example 9: If and  is in quadrant VI, find each function value. (Cont
[Solution]
d) cot
Tip: Use Reciprocal Identities.
e) csc
Tip: Use Reciprocal.

27. Example 10: If and find each function value.
Challenge!
Example 10: If and find each function value.
[Solution] Since cos > 0, then  is either in Quadrant I or VI.
Case 1) If  is in Quadrant I , then
a) sin
b) tan
c) cot

28. Example 10: If and find each function value. (Cont.)
Challenge!
Example 10: If and find each function value. (Cont.)
[Solution] Since cos > 0, then  is either in Quadrant I or VI.
Case 1) If  is in Quadrant I , then
d) sec
e) csc

29. Example 10: If and find each function value. (Cont.)
Challenge!
Example 10: If and find each function value. (Cont.)
[Solution] Since cos > 0, then  is either in Quadrant I or VI.
Case 2) If  is in Quadrant VI , then
a) sin
b) tan
c) cot

30. Example 10: If and find each function value. (Cont.)
Challenge!
Example 10: If and find each function value. (Cont.)
[Solution] Since cos > 0, then  is either in Quadrant I or VI.
Case 2) If  is in Quadrant VI , then
d) sec
e) csc

31. Assignment
P #1 – 3, 5, 7 – 13, 15 – 23 (odd), 14 – 26 (even), 29 – 33, 35