James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

A30
appendix D Trigonometry
18a
18b
18c
sin x cos y − 1 2 fsinsx 1 yd 1 sinsx 2 ydg
cos x cos y − 1 2 fcossx 1 yd 1 cossx 2 ydg
sin x sin y − 1 2 fcossx 2 yd 2 cossx 1 ydg
There are many other trigonometric identities, but those we have stated are the ones
used most often in calculus. If you forget any of the identities 13–18, remember that they
can all be deduced from Equations 12a and 12b.
EXAMPLE 6 Find all values of x in the interval f0, 2g such that sin x − sin 2x.
SOLUTIOn Using the double-angle formula (15a), we rewrite the given equation as
sin x − 2 sin x cos x or sin x s1 2 2 cos xd − 0
Therefore there are two possibilities:
sin x − 0 or 1 2 2 cos x − 0
x − 0, , 2 cos x − 1 2
x − 3 , 5 3
The given equation has five solutions: 0, y3, , 5y3, and 2.
■
Graphs of the Trigonometric Functions
The graph of the function f sxd − sin x, shown in Figure 14(a), is obtained by plotting
points for 0 < x < 2 and then using the periodic nature of the function (from Equation
11) to complete the graph. Notice that the zeros of the sine function occur at the
y
_ π 2
1
_π 0 π π 2π 5π 3π
_1 2
2
3π
2
x
(a) ƒ=sin x
y
_π
_ π 2
1
_1
0
π
2
π
3π
2
2π
5π
2
3π
x
FIGURE 14
(b) ©=cos x
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