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

Applications of Trigonometric Functions

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Applications of Trigonometric Functions


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

Amplitudea=Ymax−Ymin2PeriodP=2πbPhase 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{2\pi}{b}\\\\\hline\\ \textbf{Phase Shift}&h\\\\\hline\\ \textbf{Vertical Displacement}&k&=&\displaystyle\frac{Y_{max}+Y_{min}}{2}\\\\\hline \end{array}

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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 cosine function to describe this scenario.

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

a=38−22=18→RadiusP=150s→∴ b=2π150h=0k=38+22=20→Radius + Height above the Ground\begin{array}{rclcl} a&=&\displaystyle\frac{38-2}{2}=18&{\color{red}\rightarrow}&\footnotesize\text{Radius}\\\\ P&=&150s&{\color{red}\rightarrow}&\therefore~\footnotesize{}b=\displaystyle\frac{2\pi}{150}\\\\ h&=&0\\\\ k&=&\displaystyle\frac{38+2}{2}=20&{\color{red}\rightarrow}&\footnotesize\text{Radius + Height above the Ground}\\ \end{array}

Therefore, y=18cos⁡(2π150t)+20y=18\cos\Bigg(\displaystyle\frac{2\pi}{150}t\Bigg)+20

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

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

Practice: Applications of Trigonometric Functions

Randy goes on a rollercoaster with his girlfriend, Mira. Randy and Mira reach the top of the first hill, 12 meters above the ground, 13.2 seconds after the ride began. They reach the bottom of the hill, 4 meters above the ground, 3.2 seconds later.

How high are Randy and Mira 21.3 seconds into the ride?
The average depth of water at the end of a dock is 5 feet that varies 3 feet in both directions. The first high tide is at 3:30 AM and goes from high to low tide every 4 hours. At what times is the tide at a depth of 7 feet?
Extra Practice
Applications of Trigonometric Functions
The average depth of water at the end of a dock is 5 feet that varies 3 feet in both directions. The first high tide is at 3:30 AM and goes from high to low tide every 4 hours. At what times is the tide at a depth of 7 feet?
Applications of Trigonometric Functions
Practice: Applications of Trigonometric Functions
Randy goes on a rollercoaster with his girlfriend, Mira. Randy and Mira reach the top of the first hill, 12 meters above the ground, 13.2 seconds after the ride began. They reach the bottom of the hill, 4 meters above the ground, 3.2 seconds later.
How high are Randy and Mira 21.3 seconds into the ride?
Applications of Trigonometric Functions
A particle travels along the x-axis according to the position s(t)=3sin⁡(2t)s(t)=3\sin(2t) where tt is in seconds and ss is the x-coordinate of the particle.
a. What is the x-coordinate of the particle at t=4.2st=4.2s?
b. What is the largest value for the x-coordinate?
Applications of Trigonometric Functions
A ferris wheel has a diameter of 50 meters and completes one full rotation in 180 seconds. People board the ferris wheel at the bottom located 2 meters above the ground.
Which function describes the height, in meters, depending on the time, in seconds?