Unit 5: Graphs of the Trigonometric Functions
The unit circle unrolling into a wave, and how amplitude, period, and phase shift each reshape a sinusoid one parameter at a time.
9 topics · 35 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Unwrapping the Circle into the Sine Wave
- How it worksHow does plotting a point's height on the unit circle against the angle it has swept unroll into the sine curve?
- How it worksWalk me through tracking the y-coordinate of a point circling the unit circle as the angle grows from 0 to 2pi.
- Worked exampleFind the height of the unwrapped sine wave at an angle of 3pi/2 radians.
- Common mistakeWhy doesn't the sine wave's height ever exceed 1 even though the circle it comes from keeps turning forever?
- Going furtherWhat happens to the unwrapped wave if you track the x-coordinate of the circling point instead of the y-coordinate?
Your turn1 of 2
Find the height of the unwrapped sine wave at an angle of 5pi/5 radians.The Cosine Graph
- How it worksHow does tracking the x-coordinate of a point circling the unit circle produce the cosine curve?
- Worked exampleFind the value of cos(x) at x = 0, π/2, π, and 3pi/2 to sketch one full cycle.
- Common mistakeWhy isn't cosine just sine written with a different letter, and what actually separates their graphs?
- Ties togetherHow does the cosine graph's own shape reveal that cosine is just a phase-shifted sine wave?
Your turn1 of 2
Find the value of cos(5pi/3).Amplitude and Vertical Shift
- How it worksHow does the coefficient A in y = A·sin(x) stretch a sine wave's height without changing its period?
- Worked exampleFind the amplitude and midline of y = 3·sin(x) + 2.
- Worked exampleFind the amplitude and midline of y = -2·cos(x) - 4.
- Common mistakeWhy doesn't the amplitude of a sinusoid tell you anything about how fast it oscillates?
Your turn1 of 2
A sinusoid of the form y = A·sin(x) + D has maximum value 11 and minimum value -7. Write its equation, using the A and D these two features determine.Period and Frequency of a Sinusoid
- How it worksHow does the coefficient B in y = sin(Bx) compress or stretch a sine wave's period?
- Worked exampleFind the period of y = sin(3x) and the period of y = cos(x/2).
- Common mistakeWhy doesn't doubling the coefficient B double the period instead of cutting it in half?
Your turn1 of 2
Write a function of the form y = sin(Bx) with period 7.Phase Shift
- How it worksHow does the C in y = sin(x - C) slide a sine wave left or right without changing its shape?
- Worked exampleFind the phase shift of y = sin(2x - π/3) written in the form y = sin(B(x - C)).
- Common mistakeWhy isn't the phase shift of y = sin(2x - π/3) simply π/3?
- Common mistakeWhy doesn't a phase shift change a sinusoid's amplitude or its period at all?
Your turn1 of 2
Find the phase shift of y = sin(4x - 4pi/8) written in the form y = sin(B(x - C)).Graphing y = A sin(Bx - C) + D
- How it worksHow do amplitude, period, phase shift, and midline combine to graph one full sinusoid?
- Worked exampleFind the amplitude, period, phase shift, and midline of y = 4·sin(2x + π/2) - 1, then sketch one cycle.
- Going furtherWhat happens to the graph when the amplitude and vertical shift trade places in size?
- Common mistakeWhy does reading the phase shift wrong flip the whole graph to the opposite side of the y-axis?
Your turn1 of 2
A sinusoid has amplitude 7, period 7, phase shift 3, and midline y = -4. Write its equation in the form y = A·sin(B(x - C)) + D.The Tangent and Cotangent Graphs
- How it worksHow does tangent's period of π instead of 2pi come from sine and cosine repeating their ratio twice as often?
- Worked exampleFind the vertical asymptotes of y = tan(x) on the interval from -π to π.
- Common mistakeWhy doesn't the tangent function have an amplitude the way sine and cosine do?
Your turn1 of 2
Write the x-value of the n-th positive vertical asymptote of y = tan(3x), counting the smallest positive one as the first, as a function of n.The Secant and Cosecant Graphs
- How it worksHow does secant's graph come from taking the reciprocal of cosine at every point?
- Worked exampleFind the vertical asymptotes of y = sec(x) on the interval from 0 to 2pi.
- Common mistakeWhy can't secant or cosecant ever take a value between -1 and 1?
- Ties togetherHow does secant's graph inherit its vertical asymptotes from exactly the zeros of the cosine graph?
Your turn1 of 2
Write the x-value of the n-th positive vertical asymptote of y = sec(3x), counting the smallest positive one as the first, as a function of n.Modeling Periodic Phenomena with Sinusoids
- How it worksHow do you turn a real periodic pattern's high point, low point, and cycle length into a sinusoidal equation?
- Worked exampleFind a sinusoidal model for a Ferris wheel of diameter 40 m that completes one revolution every 6 minutes, starting at the bottom.
- Common mistakeWhy doesn't the tallest point of a sinusoidal model happen at time zero unless the problem says it does?
- Ties togetherHow does modeling a real periodic pattern pull together amplitude, period, phase shift, and midline all at once?
Your turn1 of 2
A Ferris wheel has diameter 45 m and completes one revolution every 8 minutes, starting at the bottom. Write the rider's height as a function of time t (minutes).