Verifying Trigonometric Identities Dr. Shildneck The Form of Trigonometric Verifications A Trig Verification is a proof that two expressions are equivalent. To show that two trigonometric expressions are equal, you must transform one of the expressions into the other. This transformation should be made using algebraic manipulations and by substituting (replacing) parts of the expression using the fundamental identities. The solution to the problem is not that the expressions are equal. Rather, the solution is the work in between the expressions that proves that they are equal. The final solution should be a step-by-step equivalence relation that shows
what was done to transform one side into the other. Guidelines and Hints for Writing Verifications 1. Work each side independently (you may work on each side to find the path, but your final answer has to start with one side and end with the other) 2. Start on the side you feel is more complicated. Change that, and continue to make changes, step-by-step, until you get to the other side.
3. If you get stuck, try working from the other side. You might end up in the middle. If you do, then you can just re-write that part in reverse order. 4. Look for algebraic opportunities Factor
Add Fractions (get common denominators) Square Binomials Guidelines and Hints for Writing Verifications 5. Look for ways to Substitute using the Fundamental Identities (Keep in mind what functions are on the other side) 6.
Remember, Squares are your friends! 7. Remember which functions pair up well (especially when squared) Sine and Cosine Tangent and Secant
Cotangent and Cosecant 8. If all else fails, or you cant think of anything to do, write Everything in terms of Sine and Cosine. 9. If you see addition or subtraction of fractions get a least common denominator to combine them. Guidelines and Hints for Writing Verifications
10. Dont be afraid to stop and start over. Or, if you reach a step and dont see anything else to do, back up to a previous step and try something different. 11. Always try SOMETHING! Even if it leads you down the wrong path, you learned something about the problem. 12. If you can, try to think ahead. Like in chess, if you can see what will happen a couple of moves ahead, it might help you to go the right way. 13. There are multiple correct answers to almost every identity. If it takes you 10 steps, and someone else did it differently in 5 steps, and both use correct transformations, then both of you are 100% correct! EXAMPLE 6: Verify each identity. Show all steps. sin x cos x
1 csc x sec x EXAMPLE 7: Verify each identity. Show all steps. 2 sin x 1 cos x 1 cos x EXAMPLE 2: Verify each identity. Show all steps. sin x tan( x)cos x
EXAMPLE 8: Verify each identity. Show all steps. cos x 1 sin x 2sec x 1 sin x cos x EXAMPLE 9: Verify each identity. Show all steps. 2 2
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