Trigonometric graphs

Objective. Sketch transformed sine, cosine and tangent graphs.

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Learn: the key idea

The graphs of sin θ, cos θ and tan θ follow fixed shapes with period 360° (or 2π radians) for sine and cosine, and 180° (or π radians) for tangent, with vertical asymptotes on tan θ where cos θ=0. Transformations like y=a sin(bθ)+c change amplitude (a), period (360°/b), and vertical shift (c), while y=sin(θ−d) shifts the graph horizontally by d. For y=3sin(2θ), the amplitude is 3 and the period is 360°/2=180°, so the graph completes two full cycles within 360°. Sketch key points (maxima, minima, zeros) scaled by the transformation rather than redrawing from scratch.

Worked example

State the period and amplitude of the trigonometric graph y=4cos(2θ).

1 $Identify the amplitude as the coefficient in front of cos, which is 4$
2 $Identify b=2 inside the bracket$
3 $Compute the period as 360°/b = 360°/2$
4 $State amplitude 4 and period 180°$

Answer: Amplitude 4, period 180°

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Common mistake. Students state the period of y=sin(3θ) as 3×360°=1080° instead of dividing; multiplying θ inside the function compresses the graph, so the correct period is 360°/3=120°.

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