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Grade 11 · Trigonometric periods

Trigonometric periods: Two cycles in the same window

Trigonometric periods: investigate two cycles in the same window with amplitude a = 1; angular frequency k = 2.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Trigonometric periods: Two cycles in the same window. Input radians: 0. Wave value: 0. Period: 3.142. Slope: 2Amplitude and period are different-1103.146.28yx → · labeled axes rescale to this model
The horizontal coordinate is in radians and k is positive. Playback traces input, not physical wave propagation. The graph rescales to show the selected amplitude, so numeric axis labels must be used when comparing different settings.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Input radians
0
Wave value
0
Period
3.142
Slope
2

HD animation studio

From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

Understand what you are seeing

The idea behind the motion.

Amplitude controls vertical reach while angular frequency changes the horizontal distance needed for a full repeat. The moving point traverses a fixed interval of length 2π, so the number of displayed cycles depends on k. Fractional k can show only part of a complete period. This investigation starts with Amplitude A = 1; Angular frequency k = 2. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

f(x)=A sin(kx); period=2π/k

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Predict the highest and lowest values using the amplitude alone. The starting case is “Two cycles in the same window.”

  2. STEP 2

    Follow the changing quantity

    Count cycles as the input moves from zero to 2π. Compare the count with the frequency control.

  3. STEP 3

    Explain and test the result

    Calculate 2π/k and identify matching points one period apart. The slope also depends on k, not just the amplitude.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Amplitude A = 1; Angular frequency k = 2. Pause the timeline at 40%. Given input radians = 2.513, calculate wave value, period, slope. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

Evaluate 1 sin((2)(2.513274))=-0.951057. A repeat requires kT=2π, giving T=2π/2=3.141593. Results: Wave value: -0.951; Period: 3.142; Slope: 0.618. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 11 · Trigonometry

Separate horizontal shift from angular frequency

Read a factored sinusoidal model's amplitude, period, and phase shift.