Using Trig Formulas In these sections, we will study the following topics: Using the sum and difference formulas to evaluate trigonometric functions Using the double-angle formula to evaluate trigonometric functions

2 The trig formulas that we will study in sections 8.4 and 8.5 are frequently used in calculus to rewrite trigonometric expressions in a form that helps you to simplify expressions and solve equations. In this course, we will not study the derivations of these formulas, but rather we will use them to evaluate specific trig functions and simplify trig expressions.

3

4  PLEASE NOTE THAT The same is true for the other sum and difference formulas!

5 Example Using Sine Sum/Difference Formula Solution Since we are asked to evaluate the function without the calculator, we should be able to use our special reference angles and/or the quadrantal angles. So, we want to find two special angles whose sum or difference is 75°. Let’s use the fact that 75° = ______ + ______ Evaluate sin 75° without using a calculator. Give exact answer.

6 Example Using Sine Sum/Difference Formula continued

7

8 Use a difference formula to find the exact value of cos 165° Example Using Cosine Sum/Difference Formula

9

10 Find the exact value of Example Using Tangent Sum/Difference Formula

11 Find the exact value of Solution You should recognize the form of this expression as that of the tangent sum formula, where  = _________ and  = _____________. Writing this as the tangent of a sum of angles, we get: Example Using Tangent Sum/Difference Formula

12 Write the expression as the sine of a single angle: Example Using Sine Sum/Difference Formula

13 Verify the identity: Example Using Formulas to Verify Identity

14 Verify the identity: Example Using Formulas to Verify Identity

15 Use whichever form is most convenient

16 Solution First, since  lies in Quad II, we know that: sin  is positive cos  is negative tan  is negative S A T C Example Using Double-Angle Formula

17 Solution (cont.) Secondly, we can use Pythagorean Theorem to find the missing length  Notice, we do not know the value of , nor is it necessary to find this out.

18 Solution (cont.) Finally, we will use the double angle formulas to write the three expressions. (Be careful with the signs)

19 Example Using Double-Angle Formula

21

22

23 Squared Function Formulas If we take the cosine double-angle formula and rearrange it, we obtain a new formula for sin 2  : Start with:

24 Squared Function Formulas Similarly, we can use a different cosine double-angle formula to solve for cos 2  : Start with:

25 Squared Function Formulas Finally, we can use the quotient identity to find a formula for tan 2  :

26 Here they are:

27 End of Sections