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Chapter 3: Trigonometric Identities and Equations

Difficulty Level: At Grade Created by: CK-12

Chapter Outline

Chapter Summary

Chapter Summary

Here are the identities studied in this chapter:

Quotient & Reciprocal Identities

Pythagorean Identities

Even & Odd Identities

Co-Function Identities

Sum and Difference Identities

Double Angle Identities

Half Angle Identities

Product to Sum & Sum to Product Identities

Linear Combination Formula

, where and

Review Questions

  1. Find the sine, cosine, and tangent of an angle with terminal side on .
  2. If and , find .
  3. Simplify: .
  4. Verify the identity:

For problems 5-8, find all the solutions in the interval .

  1. Solve the trigonometric equation over the interval .
  2. Solve the trigonometric equation over the interval .
  3. Solve the trigonometric equation for all real values of .

Find the exact value of:

  1. Write as a product:
  2. Simplify:
  3. Simplify:
  4. Derive a formula for .
  5. If you solve for , you would get . This new formula is used to reduce powers of cosine by substituting in the right part of the equation for . Try writing in terms of the first power of cosine.
  6. If you solve for , you would get . Similar to the new formula above, this one is used to reduce powers of sine. Try writing in terms of the first power of cosine.
  7. Rewrite in terms of the first power of cosine:

Review Answers

  1. If the terminal side is on , then the hypotenuse of this triangle would be 17 (by the Pythagorean Theorem, ). Therefore, , and .
  2. If and , then is in Quadrant II. Therefore is negative. To find the third side, we need to do the Pythagorean Theorem. So .
  3. Factor top, cancel like terms, and use the Pythagorean Theorem Identity. Note that this simplification doesn't hold true for values of that are , where is a positive integer,, since the original expression is undefined for these values of .
  4. Change secant and cosecant into terms of sine and cosine, then find a common denominator.
  5. or radians and radians
  6. Because this is , you will need to divide by 3 at the very end to get the final answer. This is why we went beyond the limit of when finding .
  7. Rewrite the equation in terms of tan by using the Pythagorean identity, . Because these factors are the same, we only need to solve one for . Where is any integer.
  8. Use the half angle formula with .
  9. Use the sine sum formula.
  10. Use the sine and cosine sum formulas.
  11. Use the sine sum formula as well as the double angle formula.
  12. Using our new formula, Now, our final answer needs to be in the first power of cosine, so we need to find a formula for . For this, we substitute everywhere there is an and the formula translates to . Now we can write in terms of the first power of cosine as follows.
  13. Using our new formula, Now, our final answer needs to be in the first power of cosine, so we need to find a formula for . For this, we substitute everywhere there is an and the formula translates to . Now we can write in terms of the first power of cosine as follows.
  14. (a) First, we use both of our new formulas, then simplify: (b) For tangent, we use the identity and then substitute in our new formulas. Now, use the formulas we derived in #18 and #19.

Texas Instruments Resources

In the CK-12 Texas Instruments Trigonometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9701.

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