The equation y =
M sin
Bt + N cos
Bt and the equation y =
A sin
(Bt +
C) are equivalent where the relationships of
A, B, C, M, and
N are as follows. The proof is direct and follows from the sum identity for sine. The following is a summary of the properties of this relationship.
M sin Bt + N cos Bt =
sin
(Bt +
C) given that Cis an angle with a point
P(M, N) on its terminal side (see Figure
1 ).
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Figure 1
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Reference graph for
y =
M sin
Bt +
N cos
Bt.
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Example 1: Convert the equation
y =
sin 3
t + 2 cos 3
t to the form
y =
A sin
(Bt + C). Find the period, frequency, amplitude, and phase shift (see Figure
2 ).
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Figure 2
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Drawing for Example 1.
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Example 2: Convert the equation
y = −sin π
t + cos π
t to the form
y =
A sin
(Bt +
C). Find the period, frequency, amplitude, and phase shift (see Figure
3 ).
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Figure 3
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Drawing for Example 2.
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Trigonometric Functions
Additional Topics
