1. H.Melikyan/12001 Verifying Trigonometric Identities Dr.Hayk Melikyan Departmen of Mathematics and CS melikyan@nccu.edu

2. H.Melikyan/12002 Basic Trigonometric Identities sin(- x ) = - sin x cos(- x ) = cosx tan( -x) = - tanx sec( -x ) = sec x csc( -x ) = - cscx cot( -x) = - cotx

3. H.Melikyan/12003 Verify the identity: sec x cot x  csc x. Solution The left side of the equation contains the more complicated expression. Thus, we work with the left side. Let us express this side of the identity in terms of sines and cosines. Perhaps this strategy will enable us to transform the left side into csc x, the expression on the right. Apply a reciprocal identity: sec x = 1 / cos x and a quotient identity: cot x = cos x / sin x. Divide both the numerator and the denominator by cos x, the common factor. Text Example

4. H.Melikyan/12004 Solution We start with the more complicated side, the left side. Factor out the greatest common factor, cos x, from each of the two terms. cos x  cos x sin 2 x  cos x(1  sin 2 x) Factor cos x from the two terms. Use a variation of sin 2 x + cos 2 x = 1. Solving for cos 2 x, we obtain cos 2 x = 1 – sin 2 x.  cos x ·  cos 2 x Multiply.  cos 3 x We worked with the left and arrived at the right side. Thus, the identity is verified. Verify the identity: cosx - cosxsin 2 x = cos 3 x Text Example

5. H.Melikyan/12005 Guidelines for Verifying Trigonometric Identities 1. Work with each side of the equation independently of the other side. Start with the more complicated side and transform it in a step  by  step fashion until it looks exactly like the other side. 2. Analyze the identity and look for opportunities to apply the fundamental identities. Rewriting the more complicated side of the equation in terms of sines and cosines is often helpful. 3. If sums or differences of fractions appear on one side, use the least common denominator and combine the fractions. 4. Don't be afraid to stop and start over again if you are not getting anywhere. Creative puzzle solvers know that strategies leading to dead ends often provide good problem  solving ideas.

6. H.Melikyan/12006 Example Verify the identity: csc(x) / cot (x) = sec (x) Solution:

7. H.Melikyan/12007 Example Verify the identity: Solution:

8. H.Melikyan/12008 Example Verify the following identity: Solution:

9. H.Melikyan/12009 Example cont. Solution:

10. H.Melikyan/120010 Sum and Difference Formulas Dr.Hayk Melikyan Departmen of Mathematics and CS melikyan@nccu.edu

11. H.Melikyan/120011 The Cosine of the Difference of Two Angles The cosine of the difference of two angles equals the cosine of the first angle times the cosine of the second angle plus the sine of the first angle times the sine of the second angle.

12. H.Melikyan/120012 Solution We know exact values for trigonometric functions of 60° and 45°. Thus, we write 15° as 60°  45° and use the difference formula for cosines. cos l5°  cos(60°  45°) cos(   )  cos  cos   sin  sin   cos 60° cos 45°  sin 60° sin 45° Substitute exact values from memory or use special triangles. Multiply. Add. Text Example v Find the exact value of cos 15°

13. H.Melikyan/120013 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 80° cos 20°  sin 80° sin 20°  cos (80°  20°)  cos 60°  1/2 cos(   )  cos  cos   sin  sin  Text Example

14. H.Melikyan/120014 Example v Find the exact value of cos(180º-30º) Solution

15. H.Melikyan/120015 Example v Verify the following identity : Solution

16. H.Melikyan/120016 Sum and Difference Formulas for Cosines and Sines

17. H.Melikyan/120017 Example v Find the exact value of sin(30º+45º) Solution

18. H.Melikyan/120018 Sum and Difference Formulas for Tangents The tangent of the sum of two angles equals the tangent of the first angle plus the tangent of the second angle divided by 1 minus their product. The tangent of the difference of two angles equals the tangent of the first angle minus the tangent of the second angle divided by 1 plus their product.

19. H.Melikyan/120019 Example v Find the exact value of tan(105º) tan(105º)=tan(60º+45º) Solution

20. H.Melikyan/120020 Example v Write the following expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. Solution