Example Identifying a Sinusoid
Determine whether the following function is or is not a sinusoid. f (x) 3cos x 5sin x
Yes, since both functions in the sum have period 2.
The function appears to be periodic.
Combining the Cosine Function with x2
Graph y cos x2 and state whether the function appears to be periodic.
Combining the Cosine Function with x2
Use the triangle to find the required ratios:
sin(sin-1 x)) x cos(sin-1 x)) 1 x2
1 csc(sin-1 x))
x 1
sec(sin-1 x)) 1 x2
x tan(sin-1 x))
1 x2
1 x2 cot(sin-1 x))
Example Composing Trig Functions with Arcsine
Compose each of the six basic trig functions with sin-1x and reduce the composite function to an algebraic expression involving no trig functions.
x
4.8
Solving Problems with Trigonometry
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Angle of Elevation, Angle of Depression
Composing Trigonometric and Inverse Trigonometric Functions
The following equations are always true whenever they are defined:
sin sin-1 x x cos cos-1 x x
No, since cos3x has period 2 / 3 and sin 5x has period 2 / 5.
Damped Oscillation
The graph of y f (x) cos bx (or y f (x)sin bx) oscillates between the graphs of y f (x) and y - f (x). When this reduces the amplitude of the wave, it is called damped oscillation. The factor f (x) is called the damping factor.
Graph y cos x2 and state whether the function appears to be periodic.
The function appears to not be periodic.
Sums That Are Sinusoisin(b(x h )) and y a cos(b(x h )), then
1
1
1
2
2
2
y y a sin(b(x h )) a cos(b(x h )) is a
1
2
1
1
2
2
sinusoid with period 2 / | b|.
Example Identifying a Sinusoid
Determine whether the following function is or is not a sinusoid. f (x) 3cos x 5sin x
tan tan-1 x x
The following equations are only true for x values in the "restricted"
domains of sin, cos, and tan:
sin1 sin x x cos-1 cos x x
tan-1 tan x x
Example Identifying a Sinusoid
Determine whether the following function is or is not a sinusoid. f (x) cos 3x sin 5x
Example Identifying a Sinusoid
Determine whether the following function is or is not a sinusoid. f (x) cos 3x sin 5x
Combining the Cosine Function with x2
Graph y cos x2 and state whether the function appears to be periodic.
Combining the Cosine Function with x2
Graph y cos x2 and state whether the function appears to be periodic.
Example Composing Trig Functions with Arcsine
Compose each of the six basic trig functions with sin-1x and reduce the composite function to an algebraic expression involving no trig functions.