5.4
Sum and Difference
Formulas
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What You Should Learn
•
Use sum and difference formulas to evaluate
trigonometric functions, verify trigonometric
identities, and solve trigonometric equations.
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Using Sum and Difference Formulas
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Using Sum and Difference Formulas
In this section and the following section, you will study the
uses of several trigonometric identities and formulas.
You need to memorize these formulas ASAP.
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Example 1 – Evaluating a Trigonometric Function
Find the exact value of (a) cos 75° and (b) sin
.
Solution:
a. Using the fact that 75° = 30° + 45° with the formula for
cos(u + v) yields
cos 75° = cos(30° + 45°)
= cos 30 cos 45 – sin 30 sin 45
=
=
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Example 1 – Solution
cont’d
Try checking this result on your calculator. You will find
that
cos 75  0.259.
b. Using the fact that
with the formula for sin (u – v) yields
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Example 1 – Solution
cont’d
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Example 3 – An Application of a Sum Formula
Write cos(arctan 1 + arccos x) as an algebraic expression.
Solution:
This expression fits the formula for cos(u + v). Angles
u = arctan 1 and v = arccos x are shown in Figures 5.21
and 5.22, respectively.
Figure 5.21
Figure 5.22
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Example 3 – An Application of a Sum Formula
cos(u + v) = cos(arctan 1) cos(arccos x)
– sin(arctan 1) sin(arccos x)
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