6.1 Sinusoidal Graphs

  1. Plot \(f(\theta)=\sin(\theta)\).
  2. What is the domain and range of sine.
  3. What is the period of sine?

  1. Plot \(f(\theta)=\cos(\theta)\).
  2. What is the domain and range of cosine?.
  3. What is the period of cosine?

Animation of sine function.

Animation of cosine function.

If \(f(x)\) is a function of \(x\), we say:

  • \(f\) is even if \(f(-x)=f(x)\).
  • \(f\) is odd if \(f(-x)=-f(x)\).

  1. Is \(f(x)=x\) even or odd?

  2. Is \(f(x)=x^2\) even or odd?

  3. Is \(f(x)=x^3\) even or odd?

  4. Is \(f(x)=\sin(x)\) even or odd?

  5. Is \(f(x)=\cos(x)\) even or odd?

  6. Simplify \(\frac{\sin(-x)}{\tan(x)}\).

Sinusoidal graphs (aka sinusoids) can be transformed in four main ways:

  • Vertical Stretch
  • Horizontal Stretch
  • Vertical Shift
  • Horizontal Shift

Vertical Stretch

A vertical stretch of a sinuoid changes the amplitude, or distance from the midline to the peak (highest value) or trough (lowest value).

  1. Sketch the graph a point rotating around a circle of radius 3. Find the equation of the y-coordinate as a function of the angle \(\theta\).
  2. What is the amplitude of the sinusoid \(f(t) = 0.2 \sin(t)\)?

Vertical Shift

A vertical stretch of a sinusoid determines the midline, the line halfway between the value of peaks and troughs.

  1. Graph the function \(f(t) = \cos(t) + 5\).
  2. If a sinusoidal function starts on the midline at point (0,3), and a period of 4, write a formula for the function.

Horizontal Stretch

A horizontal stretch of a sinuoid changes the period, or distance from one peak to the next peak, or one trough to the next trough.

  1. Sketch the graph of a point on a circle of radius 3 completes one revolution every 2 minutes. Find the equation of the y-coordinate as a function of time \(t\).

  2. What is the period of the sinusoid \(f(t) = \sin(2t)\)?

  3. What is the period of the sinusoid \(f(t) = \sin(4t)\)?

  4. What is the period of the sinusoid \(f(t) = \sin(\frac{1}{2}t)\)?

  5. What is the period of the sinusoid \(f(t) = \sin(\frac{1}{4}t)\)?

What is the period of the sinusoid \(f(t) = \sin(Bt)\)?

\[P = \frac{2\pi}{B}\]

  1. Graph the function \(f(t) = 2 \cos(\frac{\pi}{2}t)\).

Horizontal Shift

A horizontal shift changes where the sinusoid starts.

  1. Graph \(f(t) = \sin(x-\frac{\pi}{2})\).
  2. Graph \(f(t) = \sin(x+\frac{\pi}{2})\).

Note the equation \(f(t) = \sin(x-h)\) is shifted \(h\) to the right, so if \(h\) is negative (e.g. \(h=-2\)), \(f(t) = \sin(x-(-2))=\sin(x+2)\) represents a shift to the left`.

  1. Sketch the graph of \(f(t) = 3 \sin(\frac{\pi}{4}t-\frac{\pi}{4})\)

Summary

Given the equation in the form

\[f(t) = A \sin( B(t-h) ) + k\]

or \[f(t) = A \cos( B(t-h) ) + k\]

  • A is the amplitude (vertical stretch);
  • B determines the period, \(P=\frac{2\pi}{B}\);
  • k is the vertical shift, which determines the midline of the function; and
  • h is the horizontal shift.

Sketch a graph of each of the following functions and determine the amplitude, period, horizontal shift, and midline.

  1. \(y = 3\sin(\pi(x+4))+5\)
  2. \(y = 4\sin( \frac{\pi}{2}(x-3))+7\)
  3. \(y = 2\sin(\frac{1}{4}x-\frac{\pi}{2})+4\)
  4. \(y = \sin(\frac{\pi}{6}x + \pi)-3\)
  1. Find the formula of the following graph:

  1. A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meters above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function \(h(t)\) gives your height in meters above the ground \(t\) minutes after the wheel begins to turn.

    1. Find the amplitude, midline, and period of h(t).
    2. Find a formula for the height function h(t).
    3. How high are you off the ground after 5 minutes?

6.2 Graphs of Other Trig Functions

  1. Sketch a graph of \(f(x)=\tan(x)\).
  2. What is the period of \(\tan(x)\)?
  3. What is the domain and range of \(\tan(x)\)?

  1. Find the equation of the following graph.

  1. Sketch a graph of \(f(x)=\sec(x)\).
  2. What is the period of \(\sec(x)\)?
  3. What is the domain and range of \(\sec(x)\)?
  4. Recall that \(\sin(x)=\cos(\frac{\pi}{2}-x)\). Use this to show: \[\csc(x)=sec(x-\frac{\pi}{2})\]
  5. Sketch the graph or \(\csc(x)\).
  6. What is the domain, range, and period of \(\csc(x)\)?

  1. Sketch a graph of \(f(x)=2\csc(\frac{\pi}{2}x)+1\).

  2. Sketch a graph of \(f(x)=\cot(x)\).

  3. Find the period and horizontal shift of each of the following functions. Then use this information to sketch a graph of the function.

    1. \(f(x) = 2\tan(4x)\)
    2. \(h(x) = 2\sec(\frac{\pi}{4}(x+1))\)
    3. \(m(x) = 6\csc(\frac{\pi}{3}x+\pi)\)

  4. Prove the following identities.

  1. Use these identities to prove \(\tan(\theta)=-\cot(\theta-\frac{\pi}{2})\).
  2. If \(\csc(x) = 2\), find \(\csc(-x)\).
  3. If \(\tan(x) = -2\), find \(\cot(-x)\).

6.3 Inverse Trig Functions

A function \(f(x)\) is one-to-one if for every \(a,b\) where \(a\neq b\), \(f(a) \neq f(b)\).

The important thing about one-to-one functions is that they always have an inverse function \(f^{-1}(x)\), i.e., if \(f(a)=b\), then \(f^{-1}(b)=a\).

  1. Is \(f(x)=x^2\) one-to-one?

  2. Is \(f(x)=x^3\) one-to-one?

  3. Is \(\sin(x)\) one-to-one?

  4. Is \(\cos(x)\) one-to-one?

  5. Is \(\tan(x)\) one-to-one?

  1. Evaluate \(\sin^{-1}(\frac{1}{2})\).

  2. Evaluate \(\sin^{-1}(\frac{\sqrt{2}}{2})\).

  3. Evaluate \(\cos^{-1}(\frac{\sqrt{3}}{2})\).

  4. Evaluate \(\sin^{-1}(1)\).

  5. Evaluate \(\sin^{-1}(-1)\).

  6. Evaluate \(\tan^{-1}(-1)\).

  7. Evaluate \(\cos^{-1}(-1)\).

  8. Evaluate \(\cos^{-1}(\frac{1}{2})\).

  9. Evaluate \(\cos^{-1}(0.97)\) using your calculator.

  10. Evaluate \(\tan^{-1}(0.4)\) using your calculator.

  1. Using the triangle below, solve for the angle \(\theta\).

  1. Evaluate \(\cos^{-1}(\sin(\frac{13\pi}{6}))\).
  2. Evaluate \(\cos^{-1}(\sin(-\frac{11\pi}{4}))\).
  3. Evaluate \(\sin(\cos^{-1}(\frac{4}{5}))\).
  4. Evaluate \(\sin(\tan^{-1}(\frac{7}{4}))\).
  5. Find a simplified expression for \(\cos(\sin^{-1}(\frac{x}{3}))\) for \(-3 \leq x \leq 3\).
  6. Find a simplified expression for \(\sin(\tan^{-1}(4x)\) for \(-\frac{1}{4} \leq x \leq \frac{1}{4}\).

6.4 Solving Trig Equations

  1. Solve \(\sin(t)=\frac{1}{2}\) for all possible values of \(t\).

  2. A circle of radius \(5\sqrt 2\) intersects the line \(x=-5\) at two points. Find all angles \(\theta\) on the interval \(0 \leq \theta < 2\pi\) where the circle and line intersects.

  3. Solve \(\tan(t)=1\) for all possible values of \(t\)

  4. The depth of water at a dock rises and falls with the tide, following the equation \(f(t)=4\sin(\frac{\pi}{12}t)+7\) where \(t\) is measured in hours after midnight. A boat requires a depth of 9 feet to tie up at the dock. Between what times will the depth be 9 feet?

  1. Solve \(4\sin(5t)-1=1\) for all possible values of t.

  2. Find all solutions to \(\sin(\theta)=0.8\) (with use of calculator).

  3. Find all solutions to \(\sin(\theta)=-\frac{8}{9}\) on the interval \(0^{\circ}\leq \theta < 360^{\circ}\) (with use of calculator).

    1. What is the value with the largest degree value?
    2. What is the value with the smallest degree value?

  1. Find all solutions to \(\tan(\theta)=0.7\) on the interval \(0^{\circ}\leq \theta < 360^{\circ}\).

  2. Solve \(\cos(3t)=0.2\) for all solutions on two cycles.

  3. Solve \(3\sin(\pi t)=-2\) for all solutions.

  4. Solve \(3\sin(t)-2 = 2\) for all solutions.

6.5 Modeling with Trig Equations

  1. OSHA safety regulations require that the base of a ladder be placed 1 foot from the wall for every 4 feet of ladder length. Find the angle such a ladder forms with the ground.

  2. A cable that anchors the center of the London Eye Ferris wheel to the ground must be replaced. The center of the Ferris wheel is 70 meters above the ground and the second anchor on the ground is 23 meters from the base of the wheel. What is the angle from the ground up to the center of the Ferris wheel and how long is the cable?

  3. In a video game design, a map shows the location of other characters relative to the player, who is situated at the origin, and the direction they are facing. A character currently shows on the map at coordinates (-3, 5). If the player rotates counterclockwise by 20 degrees, then the objects in the map will correspondingly rotate 20 degrees clockwise. Find the new coordinates of the character.

  4. The hours of daylight in Seattle oscillate from a low of 8.5 hours in January to a high of 16 hours in July. When should you plant a garden if you want to do it during a month where there are 14 hours of daylight?

  5. An object is connected to the wall with a spring that has a natural length of 20 cm. The object is pulled back 8 cm past the natural length and released. The object oscillates 3 times per second. Find an equation for the horizontal position of the object ignoring the effects of friction. How much time during each cycle is the object more than 27 cm from the wall?

  6. A rigid rod with length 10 cm is attached to a circle of radius 4cm at point A as shown here. The point B is able to freely move along the horizontal axis, driving a piston. If the wheel rotates counterclockwise at 5 revolutions per second, find the location of point B as a function of time. When will the point B be 12 cm from the center of the circle?