1. Chapter 5 – Analytic Trigonometry 5.1 – Fundamental Identities
HW: Pg. 451 #1-8

2. Trigonometry Identities
Reciprocal Identities
csc θ= 1/sin θ
sec θ=1/cos θ
cot θ= 1/tan θ
sin θ= 1/csc θ
cos θ=1/sec θ
tan θ=1/cot θ
Quotient Identities
Tan θ = sin θ /cos θ
cot θ=cos θ/sin θ

3. Pythagorean Identities
Try to come up with an identity using the pythagorean theorem (a2 + b2 = c2) and the unit circle

4. Pythagorean Identities
Cos2θ + Sin2θ = 1
1 + tan2θ = Sec2θ
Cot2θ + 1 = csc2θ

5. Using Identities
Find sinθ and cos θ if tan θ = 5 and cos θ > 0

6. Cofunction Identities
ANGLE A:
ANGLE B:
sinA = y/r
sinB =
CosA = x/r
CosB =
tanA = y/x
tanB =
cscA = r/y
cscB =
secA = r/x
secB =
cotA = x/y
cotB =

7. Confunction Identities
Sin(π/2 – θ) = cos θ
Cos(π/2 – θ) = sin θ
Tan(π/2 – θ) = cot θ
Csc(π/2 – θ) = sec θ
Sec(π/2 – θ) = csc θ
Cot(π/2 – θ) = tan θ

8. Odd-Even Identities sin(-x) = -sinx cos(-x) = _______
tan(-x) = _______
csc(-x) = _______
sec(-x) = _______
cot(-x) = _______

9. Using Identities
If cos θ = 0.34, find sin(θ - π/2).

10. Simplifying Trigonometric Expressions
Simplify by factoring and using identities
Sin3x + sinxcos2x

11. Simplify by expanding and using identities
[(secx + 1)(secx - 1)]/sin2x

12. Simplify by combining fractions and using identities

13. Solving Trigonometric Equations
Find all values of x in the interval [0,2π) that solve cos3x / sinx = cotx

14. Solving a Trig Equation by factoring
Find all solutions to the trigonometric equation 2sin2x + sinx = 1

15. PRACTICE 5.1 – Pg. 451-452 #9-16, 23-38 (all)
Use Identities to simplify

16. 5.2 – Proving Trigonometric Identities
HW: PG. 460 #12-34e

17. Proving an Algebraic Identity
Prove the algebraic identity :
(x2-1)/(x-1) – (x2-1)/(x+1) = 2

18. General Proof Strategies I
The proof begins with the expression on one side of the identity.
The proof ends with the expression on the other side
The proof in between consists of showing a sequence of expressions, each one easily seen to be equivalent to its previous expression.

19. Proving an Identity:
tan x + cot x = sec x csc x

20. General Strategies II
Begin with the more complicated expression and work toward the less complicated expression.
If no other move suggests itself, convert the entire expression to one involving sines and cosines.
Combine fractions by combining them over a common denominator.

21. Identifying and Proving an Identity
Match the function f(x)=1/(secx - 1) + 1/(secx + 1) with (i) 2cot x csc x or (ii) 1/secx

22. Setting up a Difference of Squares
Prove the identity: cost/(1 – sint) = (1 + sint)/cost

23. General Strategies III
Use the algebraic identity (a + b)(a – b) = a2 – b2 to set up applications of the Pythagorean identities.
Always keep in mind what your goal is (expression), and favor manipulations that bring you closer to your goal.

24. Working from Both Sides
Prove the identity:
Cot2u/(1 + cscu) = (cotu)(secu – tanu)

25. Disproving Non-Identities - EXPLORATION
Prove or disprove that this is an identity: cos2x = 2cosx
Graph y=cos2x and y=2cosx in the same window. Interpret the graphs to make a conclusion about whether or not the equation is an identity.
With the help of the graph, find a value of x for which cos2x ≠ 2cosx.
Does the existence of the x value in part (2) prove that the equation is not an identity?
Graph y = cos2x and y = cos2x – sin2x in the same window. Interpret the graphs to make a conclusion about whether or not cos2x = cos2x – sin2x is an identity.
Do the graphs in part (4) prove that cos2x = cos2x – sin2x is an identity? Explain your answer.