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Wize High School Grade 12 Pre-Calculus Textbook > Trigonometric (Sinusoidal) Graphs

Equivalent Sinusoidal Functions (Radians)

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Equivalent Sinusoidal Functions (Radians)

Equivalent sinusoidal functions have many equivalent expressions because of their periodic behavior.

In order for 2 or more expressions to be equivalent, their graphs must be superimposable, which means that they lie right on top of each other.

Using Principal Angles to Write Equivalent Sinusoidal Functions

The principal angles relate to the C.A.S.T system.

Note: We may use the term reference angle instead of principal angle.

cos⁡(π−θ)=−cos⁡θPrincipal Angle in QIIsin⁡(π−θ)=sin⁡θtan⁡(π−θ)=−tan⁡θcos⁡(π+θ)=−cos⁡θPrincipal Angle in QIIIsin⁡(π+θ)=−sin⁡θtan⁡(π+θ)=tan⁡θcos⁡(2π−θ)=cos⁡θPrincipal Angle in QIVsin⁡(2π−θ)=−sin⁡θtan⁡(2π−θ)=tan⁡θ\begin{array}{|c||ccc|}\hline\\ &\cos{(\pi-\theta)}&=&-\cos{\theta}\\ \textbf{Principal Angle in QII}&\sin(\pi-\theta)&=&\sin\theta\\ &\tan(\pi-\theta)&=&-\tan{\theta}\\\\\hline\\ &\cos{(\pi+\theta)}&=&-\cos{\theta}\\ \textbf{Principal Angle in QIII}&\sin(\pi+\theta)&=&-\sin{\theta}\\ &\tan(\pi+\theta)&=&\tan{\theta}\\\\\hline\\ &\cos(2\pi-\theta)&=&\cos\theta\\ \textbf{Principal Angle in QIV}&\sin(2\pi-\theta)&=&-\sin\theta\\ &\tan(2\pi-\theta)&=&\tan\theta\\\\\hline \end{array}

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Using the Period to Write Equivalent Sinusoidal Functions


cos⁡θ=cos⁡(θ+2nπ)n∈Zsin⁡θ=sin⁡(θ+2nπ)n∈Ztan⁡θ=tan⁡(θ+nπ)n∈Z\begin{array}{rclcc} \cos\theta&=&\cos(\theta+2n\pi)&&n\in\mathbb{Z}\\\\ \sin\theta&=&\sin(\theta+2n\pi)&&n\in\mathbb{Z}\\\\ \tan\theta&=&\tan(\theta+n\pi)&&n\in\mathbb{Z} \end{array}


Categorize the Function as Even or Odd to Write Equivalent Sinusoidal Functions


cos⁡(−θ)=cos⁡θsin⁡(−θ)=−sin⁡θtan⁡(−θ)=−tan⁡θ\begin{array}{rcr} \cos(-\theta)&=&\cos\theta\\\\ \sin(-\theta)&=&-\sin\theta\\\\ \tan(-\theta)&=&-\tan\theta \end{array}
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Using Complementary Angles to Write Equivalent Sinusoidal Functions

The following are cofunction identities that describe complementary angles derived from the right triangle.

sin⁡θ=cos⁡(π2−θ)cos⁡θ=sin⁡(π2−θ)tan⁡θ=cot⁡(π2−θ)cot⁡θ=tan⁡(π2−θ)\begin{array}{rcl} \sin\theta&=&\cos\Bigg(\displaystyle\frac{\pi}{2}-\theta\Bigg)\\\\ \cos\theta&=&\sin\Bigg(\displaystyle\frac{\pi}{2}-\theta\Bigg)\\\\ \tan\theta&=&\cot\Bigg(\displaystyle\frac{\pi}{2}-\theta\Bigg)\\\\ \cot\theta&=&\tan\Bigg(\displaystyle\frac{\pi}{2}-\theta\Bigg) \end{array}


Using Horizontal Translations to Write Equivalent Sinusoidal Functions

Let's compare y=sin⁡θy=\sin{\theta} and y=cos⁡θy=\cos{\theta}.
If sin⁡θ\sin{\theta} is horizontally translated by π2\displaystyle\frac{\pi}{2} left, then we get the function y=cos⁡θy=\cos{\theta}.

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Therefore,
sin⁡(θ+π2)=cos⁡θ\boxed{\sin\Bigg(\theta+\displaystyle\frac{\pi}{2}\Bigg)=\cos\theta}

If cos⁡θ\cos{\theta} is horizontally translated by π2\displaystyle\frac{\pi}{2} right, then we get the function y=sin⁡θy=\sin{\theta}.
Therefore,
cos⁡(θ−π2)=sin⁡θ\boxed{\cos\Bigg(\theta-\displaystyle\frac{\pi}{2}\Bigg)=\sin\theta}
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Example: Equivalent Sinusoidal Functions


Use transformations to write the following as a cos⁡θ\cos\theta function and as a sin⁡θ\sin{\theta} function using the smallest possible phase shifts.


Ymax=5Yavg=2Ymin=−1Amplitude=5−(−1)2=3Vertical Displacement=5+(−1)2=2Period=π∴ b=2Phase Shift for cos⁡θ=π6Phase Shift for sin⁡θ=−π12\begin{array}{rcl} Y_{max}&=&5\\ Y_{avg}&=&2\\ Y_{min}&=&-1\\\\ \text{Amplitude}&=&\displaystyle\frac{5-(-1)}{2}&=&3\\\\ \text{Vertical Displacement}&=&\displaystyle\frac{5+(-1)}{2}&=&2\\\\ \text{Period}&=&\pi&&\therefore{}~b=2\\\\ \text{Phase Shift for }\cos\theta&=&\displaystyle\frac{\pi}{6}\\\\ \text{Phase Shift for }\sin\theta&=&-\displaystyle\frac{\pi}{12} \end{array}

Therefore,

y=3cos⁡(2(θ−π6))+2y=3\cos{\Bigg(2\Big(\theta-\displaystyle\frac{\pi}{6}\Big)\Bigg)}+2


y=3sin⁡(2(θ+π12))+2y=3\sin{\Bigg(2\Big(\theta+\displaystyle\frac{\pi}{12}\Big)\Bigg)}+2
Write an expression that is equivalent to each of the following using the cofunction identity:

  1. cos⁡(π7)\cos{\Bigg(\displaystyle\frac{\pi}{7}\Bigg)}
  2. sin⁡(2π5)\sin{\Bigg(\displaystyle\frac{2\pi}{5}\Bigg)}
  3. tan⁡(π12)\tan{\Bigg(\displaystyle\frac{\pi}{12}\Bigg)}

Practice: Equivalent Sinusoidal Functions


Choose which of the following are functions that describe the following graph:


Practice: Equivalent Sinusoidal Functions

True or False?

−2cos⁡(3(θ−π9))+1=−2sin⁡(3(θ−5π18))+1-2\cos\Bigg(3\Big(\theta-\displaystyle\frac{\pi}{9}\Big)\Bigg)+1=-2\sin\Bigg(3\Big(\theta-\displaystyle\frac{5\pi}{18}\Big)\Bigg)+1