Grade 11 · Trigonometric periods · 61 of 600
Trigonometric periods · Amplitude A=0.5; Angular frequency k=1
Evaluate at x=6π/5 radians, keeping π exact during calculation. Find the signal value, derivative with respect to x, and period. Givens: Amplitude A=0.5; Angular frequency k=1.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
The signal is y=0.5 sin(1x), with x in radians. Evaluate at x=6π/5 radians, keeping π exact during calculation. Find the signal value, derivative with respect to x, and period.
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A starting point
The chain rule gives y′=Ak cos(kx). Solve kT=2π to obtain the period.
Work through the reasoning
Step 1
Identify the model and target
The signal is y=0.5 sin(1x), with x in radians. Evaluate at x=6π/5 radians, keeping π exact during calculation. The governing relation is f(x)=A sin(kx); period=2π/k. The chain rule gives y′=Ak cos(kx). Solve kT=2π to obtain the period.
Step 2
Substitute and calculate
Evaluate 0.5 sin((1)(3.769911))=-0.293893. A repeat requires kT=2π, giving T=2π/1=6.283185.
Step 3
Check the mathematical meaning
The amplitude bound is |y|≤0.5; the computed value is -0.293893. Multiplying the period 6.283185 by k=1 gives approximately 2π. Input radians: 3.77; Wave value: -0.294; Period: 6.283; Slope: -0.405. Decimal values are rounded, so use unrounded intermediate values.
The answer
Evaluate 0.5 sin((1)(3.769911))=-0.293893. A repeat requires kT=2π, giving T=2π/1=6.283185. The amplitude bound is |y|≤0.5; the computed value is -0.293893. Multiplying the period 6.283185 by k=1 gives approximately 2π. Animation check: Input radians: 3.77; Wave value: -0.294; Period: 6.283; Slope: -0.405. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
PausedHD animation studio
From experiment to screen.
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The mathematical idea
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. The signal is y=0.5 sin(1x), with x in radians. Evaluate at x=6π/5 radians, keeping π exact during calculation. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
f(x)=A sin(kx); period=2π/k
03 · Reflect and transfer
Explain what changes and why.
Double the frequency without changing amplitude. Compare the period and the largest slope magnitude.
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.