1. Remember an identity is an equation that is true for all defined values of a variable. We are going to use the identities that we have already established to "prove" or establish other identities.

2. RECIPROCAL IDENTITIES QUOTIENT IDENTITIES PYTHAGOREAN IDENTITIES EVEN-ODD IDENTITIES

3. COFUNCION IDENTITIES

4. Get common denominators If you have squared functions look for Pythagorean Identities Work on the more complex side first If you have a denominator of (1 + trig function) try multiplying top and bottom by conjugate and use Pythagorean Identity When all else fails write everything in terms of sines and cosines using reciprocal and quotient identities Hints for Establishing Identities

5. Using the identities you now know, find the trig value: 1.) If cos θ = 3/4, find sec θ 2.) If cos θ = 3/5, find csc θ.

6. 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 sin or/and cos and then simplify. Let’s see an example of this: substitute using each identity simplify

7. sec x csc x sec x csc x 1 sin x 1 cos x 1 sinx 1 = x = cos x = tan x Simplifying with trig identities

8. Example Simplify the trig expression: Solution:

9. Example Simplify: = cot x (csc 2 x - 1) = cot x (cot 2 x) = cot 3 x Factor out cot x Use Pythagorean identity Simplify

10. Example Simplify: Use quotient identity Simplify fraction with LCD Simplify numerator = sin x (sin x) + cos x cos x = sin 2 x + (cos x) cos x = sin 2 x + cos 2 x cos x = 1 cos x = sec x Use Pythagorean identity Use reciprocal identity

11. Example Combine fraction Simplify the numerator Use Pythagorean identity Use Reciprocal Identity

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

13. Simplifying Trigonometric Expressions c) (1 + tan x) 2 - 2 sin x sec xd)

14. Simplify each expression.

15. Prove tan(x) cos(x) = sin(x)

16. Example Verify the identity :

17. Prove tan 2 (x) = sin 2 (x) cos -2 (x)

18. Prove

20. Example: Verifying identities Verify that 1 – cos 2x sec 2 x = tan 2 x is an identity.

21. Trigonometric Identities Summation & Difference Formulas

22. find the exact value of In order to answer this question, we need to find two of the angles that we know to either add together or subtract from each other that will get us the angle π/12. Let’s start by looking at the angles that we know : continued on next slide

23. We have several choices of angles that we can subtract from each other to get π/12. We will pick the smallest two such angles: continued on next slide Now we will use the difference formula for the sine function to calculate the exact value.

24. For the formula a will be continued on next slide and b will be

26. Simplify In order to answer this question, we need to use the sine formula for the sum of two angles. continued on next slide For the formula a will be and b will be

28. Summary of Double-Angle Formulas

29. Trigonometric Identities Half Angle Formulas The quadrant of determines the sign.

30.   4 5 -3 Use triangle to find values. Let's draw a picture.

31.   4 5 -3 Use triangle to find cosine value. If  is in quadrant II then half  would be in quadrant I where sine is positive

32. Ex. Use the following to find sin2 , cos2  and tan2  5 13 -12

33. Ex Use the following to find sin2 , cos2  and tan2  5 13 -12

34. Ex Use the following to find sin2 , cos2  and tan2  5 13 -12

35. and  terminates in the first quadrant, find the exact value of sin2 , cos2  and tan 2 . 5 Ex

36. We could find sin 15° using the half angle formula. Since 15° is half of 30° we could use this formula if  = 30° 30° 15° is in first quadrant and sine is positive there so we want the +

37. Example: Finding an Exact Value Find the exact value of sin (  /8).