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Wize High School Grade 11 Math Textbook > Sinusoidal Functions (degrees)

Applications of Trigonometric Functions

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Applications of Trigonometric Functions

Let's use our knowledge to model things that occur cyclically using sinusoidal functions.

We'll be presented with graphs or scenarios, and we'll use the following formulas to write out the correct transformed function.

Formulas

If y=asin⁡(b(x−h))+ky=a\sin(b(x-h))+k or y=acos⁡(b(x−h))+ky=a\cos(b(x-h))+k, then:

Amplitudea=ymax−ymin2PeriodP=360°∣b∣Phase ShifthVertical Displacementk=ymax+ymin2\begin{array}{|c|rcl|}\hline\\ \textbf{Amplitude}&a&=&\displaystyle\frac{y_{max}-y_{min}}{2}\\\\\hline\\ \textbf{Period}&P&=&\displaystyle\frac{360\degree}{|b|}\\\\\hline\\ \textbf{Phase Shift}&h\\\\\hline\\ \textbf{Vertical Displacement}&k&=&\displaystyle\frac{y_{max}+y_{min}}{2}\\\\\hline \end{array}

Wize Tip
If the scenario starts at a maximum (or minimum point), it is often easier to use cos⁡x\cos x.

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Example: Applications of Trigonometric Functions

A Ferris wheel is 2 meters above the ground. It has a radius of 18 meters and takes 150 seconds to make one full rotation. If the riders load at the bottom of the ride, write a function that models this scenario.

Since riders start at the bottom (yminy_{min}), this is easiest to write with a vertically reflected cosine: −cos⁡x-\cos x
The minimum height is the bottom of the ferris wheel: ymin=2y_{min} = 2
The maximum height is the bottom of the ferris wheel PLUS twice the radius: ymax=2+(18×2)=38y_{max} = 2+(18\times2) = 38

Let y=acos⁡(b(x−h))+ky=a\cos(b(x-h))+k.

a=ymax−ymin2=38−22=18→RadiusP=150s→∴ b=360°150h=0k=ymax+ymin2=38+22=20→Center of wheel: radius + height above ground\begin{array}{rclcl} a&=&\dfrac{y_{max}-y_{min}}{2} =\displaystyle\frac{38-2}{2}=18&{\color{red}\rightarrow}&\text{Radius}\\\\ P &=&150s&{\color{red}\rightarrow}&\therefore~b=\displaystyle\frac{360\degree}{150}\\\\ h&=&0\\\\ k&=&\dfrac{y_{max}+y_{min}}{2} =\displaystyle\frac{38+2}{2}=20&{\color{red}\rightarrow}&\text{Center of wheel: radius + height above ground}\\ \end{array}

Therefore, y=−18cos⁡(360°150t)+20\boxed{y=-18\cos\Bigg(\displaystyle\frac{360\degree}{150}t\Bigg)+20}

Practice: Applications of Trigonometric Functions

A particle travels along the y-axis according to s(t)=3sin⁡(2t)s(t)=3\sin(2t) where tt is in seconds and s(t)s(t) is the y-coordinate of the particle.

a) What is the y-coordinate of the particle at t=4.2st=4.2s?
b) What is the largest value for the y-coordinate?

Practice: Applications of Trigonometric Functions

Randy goes on a rollercoaster with many identical hills. He reaches the top of the first hill, 12 meters above the ground, after 13.2 seconds. He reaches the bottom of the hill, 4 meters above the ground, after another 3.2 seconds.

How high is Randy 21.3 seconds into the ride?

Practice: Applications of Trigonometric Functions

The average depth of water at the end of a dock is 5 feet, and the depth varies by 3 feet in both directions.
The first high tide is at 3:30AM and goes from high to low tide every 4 hours.
At what times is the tide at a depth of 7 feet?