Chapter 7: Circular Functions
Chapter 7 Summary and Review
Key Concepts
- Changes to the amplitude, period, and midline of the basic sine and cosine graphs are called transformations. Changing the midline shifts the graph vertically, changing the amplitude stretches or compresses the graph vertically, and changing the period stretches or compresses the graph horizontally.
- The order in which we apply transformations to a function makes a difference in the graph.
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Amplitude, Period, and Midline of Sinusoidal Functions.
- The graph of
[latex]y=A\cos x ~~~~{or}~~~~ y=A\sin x[/latex]
has amplitude [latex]\lvert A \rvert{.}[/latex]
- The graph of
[latex]y=\cos Bx ~~~~{or}~~~~ y=\sin Bx[/latex]
has period [latex]\dfrac{2\pi}{B}{.}[/latex]
- The graph of
[latex]y=k+\cos x ~~~~{or}~~~~ y=k+\sin x[/latex]
has midline [latex]y=k{.}[/latex]
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Horizontal Shifts.
The graphs of
[latex]y = \sin (x-h)~~~~ {and}~~~~ t = \cos (x-h)[/latex]
are shifted horizontally compared to the graphs of [latex]y = \sin x[/latex] and [latex]y = \cos x{.}[/latex]
- If [latex]h \gt 0{,}[/latex] the graph is shifted to the right.
- If [latex]h \lt 0{,}[/latex] the graph is shifted to the left.
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Standard Form for Sinusoidal Functions.
The graphs of the functions
[latex]y = A\sin B(x-h) + k~~~~ {and}~~~~ y = A\cos B(x-h) + k[/latex]
are transformations of the sine and cosine graphs.
- The amplitude is [latex]\lvert {A} \rvert{.}[/latex]
- The midline is [latex]y = k{.}[/latex]
- The period is [latex]\dfrac{2\pi}{\lvert {B} \rvert },~~B \not= 0{.}[/latex]
- The horizontal shift is [latex]h[/latex] units to the right if [latex]h[/latex] is positive, and [latex]h[/latex] units to the left if [latex]h[/latex] is negative.
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Solutions of Trigonometric Equations.
- The equation [latex]\cos \theta = k, ~~ -1\lt k \lt 1[/latex] has two solutions between [latex]0[/latex] and [latex]2\pi{:}[/latex]
[latex]\theta_{1}=\cos^{-1}(k)~~ {and}~~ \theta_{2}=2\pi -\theta_{1}[/latex]
- The equation [latex]\sin \theta = k, ~~ -1\lt k \lt 1[/latex] has two solutions between [latex]0[/latex] and [latex]2\pi{:}[/latex]
[latex]{If}~k \gt 0:~~ \theta_{1} =\sin^{-1}(k)~~ {and}~~ \theta_{2}=\pi -\theta_{1}[/latex][latex]{If}~k \lt 0:~~ \theta_{1}=\sin^{-1}(k) + 2\pi ~~ {and}~~ \theta_{2}=\pi -\sin^{-1}(k)[/latex]
- The equation [latex]\tan \theta = k[/latex] has two solutions between [latex]0[/latex] and [latex]2\pi{:}[/latex]
[latex]{If}~k \gt 0:~~ \theta_{1}=\tan^{-1}(k)~~ {and}~~ \theta_{2}=\pi +\theta_{1}[/latex][latex]{If}~k \lt 0:~~ \theta_{1}=\tan^{-1}(k) + \pi ~~ {and}~~ \theta_{2}=\pi +\theta_{1}[/latex]
- If [latex]n[/latex] is a positive integer, the equations [latex]\sin n\theta = k[/latex] and [latex]\cos n\theta = k[/latex] each have [latex]2n[/latex] solutions between [latex]0[/latex] and [latex]2\pi[/latex] for [latex]-1\lt k \lt 1{.}[/latex]
- The equation [latex]\tan n\theta = k[/latex] has one solution in each cycle of the graph.
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Using a Substitution to Solve Trigonometric Equations.
To solve the equation [latex]\sin (Bx+C)=k[/latex] or [latex]\cos (Bx+C)=k{:}[/latex]
- Substitute [latex]\theta = Bx+C[/latex] and find two solutions for [latex]\sin \theta = k[/latex] or [latex]\cos \theta = k{.}[/latex]
- Replace [latex]\theta[/latex] by [latex]Bx+C[/latex] in each solution and solve for [latex]x{.}[/latex]
- Find the other solutions by adding multiples of [latex]\dfrac{2\pi}{B}[/latex] to the first two solutions.
To solve the equation [latex]\tan (Bx+C)=k{:}[/latex]
- Substitute [latex]\theta = Bx+C[/latex] and find one solution for [latex]\tan \theta = k{.}[/latex]
- Replace [latex]\theta[/latex] by [latex]Bx+C[/latex] and solve for [latex]x{.}[/latex]
- Find the other solutions by adding multiples of [latex]\dfrac{\pi}{B}[/latex] to the first solution.