Sum and Difference Formulas Exploration

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3
2
1
−1
−2
−3
y
FIGURE 5.8
π
2
π 2π
π
( (
4
(
π
4
= sin x + + sin x − (y + 1
x
Example 7
Solving a Trigonometric Equation
Find all solutions of sinx sinx in the interval 0, 2.
Solution
Using sum and difference formulas, rewrite the equation as
sin x cos
4
So, the only solutions in the interval 0, 2 are
5
x
4
and
You can confirm this graphically by sketching the graph of
y sinx
1 for 0 ≤ x < 2,
as shown in Figure 5.8. From the graph you can see that the x-intercepts
are 54
and 74.
Now try Exercise 69.
The next example was taken from calculus. It is used to derive the derivative
of the sine function.
Example 8
Verify that
sinx h sin x
h
where h 0.
Solution
cos x sin
4
4
4
x 7
4 .
sin x
An Application from Calculus
cos x
4
sin h
h
Using the formula for sinu v, you have
Section 5.4 Sum and Difference Formulas 403
4 1
sin x cos cos x sin
4 4 1
2 sin x cos
4 1
2sin x 2
2 1
sin x 2
2 .
sin x 1
2
1 cos h
sin x h
sin h
1 cos h
cos x h sin x h
Now try Exercise 91.
.
sinx h sin x sin x cos h cos x sin h sin x
h
h
cos x sin h sin x1 cos h
h