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Finding Exact Trigonometric Values Using Sum and Difference Formulas

Convert angles to sum or difference of 30, 45, and 60 degrees to solve.

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Finding Exact Trig Values using Sum and Difference Formulas

You measure an angle with your protractor to be . How could you find the exact sine of this angle without using a calculator?

Sum and Difference Formulas

You know that etc... from the special right triangles. In this concept, we will learn how to find the exact values of the trig functions for angles other than these multiples of and . Using the Sum and Difference Formulas, we can find these exact trig values.

Sum and Difference Formulas

Let's find the following exact values using the Sum and Difference Formulas.

This is an example of where we can use the sine sum formula from above, , where and .

In general, and similar statements can be made for the other sum and difference formulas.

For this problem, we could use either the sum or difference cosine formula, or . Let’s use the sum formula.

This angle is the difference between and .

This angle is also the same as . You could have also used this value and done and arrived at the same answer.

Examples

Example 1

Earlier, you were asked to find the exact value of  without using the calculator. 

We can use the sine sum formula, , where and .

Example 2

Find the exact value of .

Example 3

Find the exact value of .


Review

Find the exact value of the following trig functions.

  1. Now, use from #1, and find . Do you arrive at the same answer? Why or why not?
  2. Using from #7, find . What is another way you could find ?
  3. Describe any patterns you see between the sine, cosine, and tangent of these “new” angles.
  4. Using your calculator, find the . Now, use the sum formula and your calculator to find the using and .
  5. Use the sine difference formula to find with any two angles you choose. Do you arrive at the same answer? Why or why not?
  6. Challenge Using and , show that .

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 14.12. 

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