1. (x, y)
(- x, y)
(- x, - y)
(x, - y)

2. Sect 5.1 Verifying Trig identities
Reciprocal
Co-function
Pythagorean
Even/Odd
Quotient

3. If and is in quadrant II, find each function value.
S A
T C
If and is in quadrant II, find each function value.
Negative answer.
Positive answer.
Positive answer.
What Trig. Identity has tan and sec?
What Trig. Identity has tan and cot?
What Trig. Identity has tan and sin?

4. Write cos(x) in terms of tan(x).
Secant has a relationship with both tangent and cosine.
Rationalize the denominator.

5. Write in terms of sin(x) and cos(x), and simplify the expression so
that no quotients appear.

6. Sect 5.2 Verifying Trig identities
Guidelines to follow.
1. Work with one side of the equation at a time. It is often better to work on the most complicated.
Look for opportunities to factor, add fractions, square binomials or multiply a binomial by it’s conjugate to create a monomial.
Look to use fundamental identities. Look to see what trig functions are in the answer.
Convert everything to sines and cosines
5. Always try something!

7. Sect 5.2 Verifying Trig identities
Work on the right side first.
Verify.
Distribute the cosecant.
Rewrite to sine and cosine.
Simplify the fractions.
Quotient Identity for cotangent.

8. Sect 5.2 Verifying Trig identities
Work on the left side first.
Verify.
Pythagorean Identity
1 + cot2x = csc2x
Rewrite to sine and cosine.
Simplify the fractions by canceling .
Reciprocal Identity for secant.

9. Sect 5.2 Verifying Trig identities
Work on the left side first.
Verify.
Rewrite the fraction as subtraction of two fractions with the same denominators.
Rewrite to sine and cosine.
Simplify the fractions by multiplying by the reciprocals and cancel.
Reciprocal Identity for secant and cosecant.

10. Sect 5.2 Verifying Trig identities
Work on the left side first.
Verify.
Pythagorean Identity
1 + tan2x = sec2x
tan2x = sec2x – 1
Rewrite to sine and cosine.
Rewrite as multiplication.
Cancel and Simplify.

11. Sect 5.2 Verifying Trig identities
Work on the right side first. Two terms need to be condensed to one term. Find LCD and combine the fractions.
Verify.
Pythagorean Identity
sin2x + cos2x = 1
cos2x = 1 – sin2x
Reciprocal of cosine.

12. Sect 5.2 Verifying Trig identities
Work on the right side first. Pythagorean Identities.
Verify.
sin2x + cos2x = 1
cos2x – 1 = – sin2x
1 + tan2x = sec2x
Convert to cosine.
Multiply.

13. Sect 5.2 Verifying Trig identities
Work on the left side first. Try to combine the two terms into one.
Verify.
Convert to sine and cosine.
Pythagorean Identity
sin2x + cos2x = 1
Rewrite as two fractions multiplied together.
Reciprocals.

14. Sect 5.2 Verifying Trig identities
Work on the right side first. Two terms need to be condensed to one term.
Verify.
Convert to sine and cosine.
Combine.
When working with binomials, try multiplying by the conjugate to create differences of squares which will incorporate the Pythagorean Identities.
Pythagorean Identity
sin2x + cos2x = 1
cos2x = 1 – sin2x
Cancel cosine.

15. Sect 5.2 Verifying Trig identities
Work on the left side first. Pythagorean Identity and convert to sine and cosine.
Verify.
1 + cot2x = csc2x
cot2x = csc2x – 1
csc2x – 1 is Diff. of Squares.
Factor.
Cancel (csc x + 1)
Convert to sine.
Combine to one term.

16. Sect 5.3 Sum and Difference Formulas
Using Distance Formula
A – B
A – B B
Dist. from (cos(A-B), sin(A-B)) to (1,0)
= Dist. from (cosA, sinA) to (cosB,sinB)
F.O.I.L F.O.I.L F.O.I.L.
Pythagorean Identity
Pythagorean Identity
Pythagorean Identity
Subtract by 2.
– 2
– 2
Divide by –2.
The Cosine of the Difference of Two Angles

17. The Cosine of the Difference of Two Angles
Substitute (-B) for B in the formula to make the Cosine of the Sum of Two Angle.
cos (– B) = cos (B)
sin (– B) = – sin (B)
The Cosine of the Sum of Two Angles

18. To make the Sine of the Sum & Difference of Two Angles we will need the Cofunction Identities for Sine and Cosine.
Start with
Substitute (-B) for B in the formula to make the Sine of the Sum of Two Angle.
cos (– B) = cos (B)
sin (– B) = – sin (B)

19. To make the Tangent of the Sum & Difference of Two Angles we will need the Quotient Identities for Tangent.
This is what we need divide by all the factors.
cos (A)
cos (B)
cos (A)
cos (B)
Tricky manipulation: We want this fraction to have tangents in the formula. Need to divide by the same factor in both the top and bottom to make tangents.
Start with where we need to divide by cosine.

20. This is what we need divide by all the factors.
cos (A)
cos (B)

21. Find the exact value of
Use the special right triangle angles, 30o, 45o, and 60o. We may need to use multiples of these angles.

22. Find the exact value of
Use the special right triangle angles, 30o, 45o, and 60o. We may need to use multiples of these angles.

23. Suppose that for a Q2 angle and for a Q1 angle .
Find the exact value of each of the following.
A. B. C. D.

24. Find the exact value of
Use the special right triangle angles, 30o, 45o, and 60o. We may need to use multiples of these angles.

25. Find the exact value of
Use the special right triangle angles, 30o, 45o, and 60o. We may need to use multiples of these angles.

26. Find the exact value of . Another approach.
Use the special right triangle angles, 30o, 45o, and 60o. We may need to use multiples of these angles.

27. Find the exact value of

28. => Sect 5.5 Dble Angle, Power Reducing, and Half Angle Formulas
Double Angle Formulas: Revise the Sum of Sin, Cos, & Tan Formulas
Substitute A in for B.
=>

29. Quadrant 4.
Find given and

30. Find the values of the six trigonometric functions of if and .
Quadrant 2.
Find the values of the six trigonometric functions of if and
Choose one of the double angle identities to find a value for sine or cosine.
SOH-CAH-TOA
Substitute in 4/5.
Subtract by 1.
Divide by -2.
Square root both sides, but the answer will be positive, since we are Q2.

31. Verify.
Work on the left side first. Convert to sine and cosine with Quotient Identity.
Double angle identity. 2sin(x) cos(x) = sin(2x)
Cancel
Rewrite the double angle formula.
2cos2x – 1 = cos(2x)
2cos2x = 1 + cos(2x)

33. ( ) ( ) ( ) ( ) Find an identity for Substitute Dble angle Identity.
( )
( )
Substitute Dble angle Identity.
( )
( )
Pythagorean Identity, rewrite with all cosines.
Distribute

34. Product to Sum & Sum to Product Formulas
How to create the Product to Sum Formulas. Add and subtract Sum and Difference formulas for Sine and Cosine.

35. Product to Sum Formulas
Sum to Product Formulas
The reason we choose these two fractions for A and B is because we need two values that add up to x and two values that subtract to be y.

36. Product to Sum Formulas
Sum to Product Formulas

37. Rewrite as a sum or difference of two functions
Rewrite using sums to product identity.

38. Half Angle Formulas
The + symbol in each formula DOES NOT mean there are 2 answers, instead it indicates that you must determine the sign of the trigonometric functions based on which quadrant the half angle falls in.

40. Find the exact value for .
S A
T C
Verify the identity.