1. Basic Trigonometric Identities and Equations
By the end of this chapter, you should be able to:
-identify non-permissible values for trigonometric expressions
-show that a trigonometric identity is true for all permissible values of the variable by using algebra (not just by substituting numbers in for the variable or by graphing)
-Use trigonometric identities to simplify more complicated trigonometric expressions
-solve trigonometric equations algebraically
-find exact values for given trigonometric expressions

2. Trigonometric Identities
Quotient Identities
Reciprocal Identities
Pythagorean Identities
sin2q + cos2q = 1
tan2q + 1 = sec2q
cot2q + 1 = csc2q
sin2q = 1 - cos2q
tan2q = sec2q - 1
cot2q = csc2q - 1
cos2q = 1 - sin2q
5.4.3

3. Do you remember the Unit Circle?
Where did our pythagorean identities come from??
Do you remember the Unit Circle?
What is the equation for the unit circle?
x2 + y2 = 1
What does x = ? What does y = ?
(in terms of trig functions)
sin2θ + cos2θ = 1
Pythagorean Identity!

4. Take the Pythagorean Identity and discover a new one!
Hint: Try dividing everything by cos2θ
sin2θ + cos2θ =
cos2θ cos2θ cos2θ
tan2θ = sec2θ
Quotient
Identity
Reciprocal
Identity
another Pythagorean Identity

5. Take the Pythagorean Identity and discover a new one!
Hint: Try dividing everything by sin2θ
sin2θ + cos2θ =
sin2θ sin2θ sin2θ
cot2θ = csc2θ
Quotient
Identity
Reciprocal
Identity
a third Pythagorean Identity

6. Using the identities you now know, find the trig value.
1.) If cosθ = 3/4, find secθ 2.) If cosθ = 3/5, find cscθ.

7. Simplifying Trigonometric Expressions
Identities can be used to simplify trigonometric expressions.
Simplify. b) a)
5.4.5

8. Simplifing Trigonometric Expressions
c) (1 + tan x)2 - 2 sin x sec x d)

9. Simplify each expression.

10. Simplifying trig Identity
Example1: simplify tanxcosx
sin x
cos x
tanx cosx
tanxcosx = sin x

11. Simplifying trig Identity
sec x
csc x
Example2: simplify 1
cos x 1
cos x
sinx = x
sec x
csc x 1
sin x =
sin x
cos x
= tan x

12. Simplifying trig Identity
cos2x - sin2x
cos x
Example2: simplify
= sec x
cos2x - sin2x
cos x
cos2x - sin2x 1

13. Example Simplify: = cot x (csc2 x - 1) Factor out cot x
= cot x (cot2 x)
Use pythagorean identity
= cot3 x
Simplify

14. Example Simplify: = sin x (sin x) + cos x Use quotient identity cos x
Simplify fraction with LCD
= sin2 x + (cos x)
cos x
= sin2 x + cos2x
cos x
Simplify numerator =
cos x
Use pythagorean identity
= sec x
Use reciprocal identity

15. Your Turn! Combine fraction Simplify the numerator
Use pythagorean identity
Use Reciprocal Identity

16. One way to use identities is to simplify expressions involving trigonometric functions. Often a good strategy for doing this is to write all trig functions in terms of sines and cosines and then simplify. Let’s see an example of this:
substitute using each identity
simplify

17. Another way to use identities is to write one function in terms of another function. Let’s see an example of this:
This expression involves both sine and cosine. The Fundamental Identity makes a connection between sine and cosine so we can use that and solve for cosine squared and substitute.

18. Sum and Difference Identities

19. Combined Sum and Difference Formulas

20. These identities are useful to find exact answers for non-special angles
Example Find the exact value of the following.
cos 15°
cos
(or 60° – 45°)

21. Example Find the exact value of the following.
sin 75°
tan
sin 40° cos 160° – cos 40° sin 160°
Solution
(a)

22. (b)
(c) sin 40°cos 160° – cos 40°sin 160°
=sin(40°-160°)
= sin(–120°)

23. Example
Find the exact value of ( cos 80° cos 20° + sin 80° sin 20°) .
Solution The given expression is the right side of the formula for cos( - ) with  = 80° and  = 20°.
cos( -) = cos  cos  + sin  sin 
cos 80° cos 20° + sin 80° sin 20° = cos (80° - 20°)
= cos 60° = 1/2

24. Example
Write the following expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression.
Solution:

25. DOUBLE-ANGLE IDENTITIES

26. If we want to know a formula for we could use the sum formula.
we can trade these places
This is called the double angle formula for sine since it tells you the sine of double 

27. Let's try the same thing for
This is the double angle formula for cosine but by substiuting some identities we can express it in a couple other ways.

28. Double-angle Formula for Tangent

29. Summary of Double-Angle Formulas

30. Your Turn: Simplify an Expression
Simplify cot x cos x + sin x.
Click for answer.
Page 189

31. Your Turn: Cosine Sum and Difference Identities
Find the exact value of cos 75°.
Click for answer.
Page 198

32. Your Turn: Sine Sum and Difference Identities
Find the exact value of
Click for answer.

33. Your Turn: Double-Angle Identities
If , find sin 2x given sin x < 0.
Click for answer.

34. Your Turn: Double-Angle Identities