Verifying Trigonometric Identities

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

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

3. Text Example 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/cosx and a quotient identity: cot x = cosx/sinx. Divide both the numerator and the denominator by cos x, the common factor.

4. 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 sin2x= cos x(1 - sin2x) Factor cos x from the two terms. Use a variation of sin2 x + cos2 x = 1. Solving for cos2 x, we obtain cos2 x = 1 – sin2 x. = cos x ·cos2x = cos3x Multiply. Text Example Verify the identity: cosx - cosxsin2x = cos3x We worked with the left and arrived at the right side. Thus, the identity is verified.

5. Guidelines for Verifying Trigonometric Identities • 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. • 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. • If sums or differences of fractions appear on one side, use the least common denominator and combine the fractions. • 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. Example Verify the identity: csc(x) / cot (x) = sec (x) Solution:

7. Example Verify the identity: Solution:

8. Example Verify the following identity: Solution:

9. Example cont. Solution:

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

11. 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. 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 60° cos 45° + sin 60° sin 45° cos(-) = cos  cos + sin  sin  Text Example • Find the exact value of cos 15° Substitute exact values from memory or use special triangles. Multiply. Add.

13. cos(-) = cos  cos + sin  sin  Text 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 80° cos 20° + sin 80° sin 20° = cos (80° - 20°) = cos 60° = 1/2

14. Example • Find the exact value of cos(180º-30º) Solution

15. Example • Verify the following identity: Solution

16. Sum and Difference Formulas for Cosines and Sines

17. Example Solution • Find the exact value of sin(30º+45º)

18. 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. Example Solution • Find the exact value of tan(105º) • tan(105º)=tan(60º+45º)

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