Trigonometric graphs

Objective. Sketch and interpret sine, cosine and tangent graphs.

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2. Worked example
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Learn: the key idea

Trigonometric graphs plot sine, cosine and tangent functions against angle, revealing their period, amplitude and symmetry: sin x and cos x oscillate between −1 and 1 with a period of 360°, while tan x repeats every 180° and has vertical asymptotes where cos x=0. The sine graph passes through the origin, while the cosine graph starts at its maximum value of 1 when x=0. Transformations like y=a sin(x) change the amplitude, and y=sin(x+c) shifts the graph horizontally by c. For example, y=2cos(x) oscillates between −2 and 2, reaching its maximum of 2 at x=0°.

Worked example

State the period and the maximum value of the graph y = 3sin(x).

1 $Recognise that sin(x) has period 360° regardless of the coefficient 3$
2 $The coefficient 3 affects the amplitude, not the period$
3 $Since sin(x) ranges from -1 to 1, 3sin(x) ranges from -3 to 3$
4 $Identify the maximum value as 3$

Answer: Period 360°; maximum value 3

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Common mistake. Students often think tan x is continuous everywhere like sine and cosine, but it has vertical asymptotes at x=90°, 270°, etc., where the function is undefined because cos x=0 there, causing the graph to shoot off to infinity.

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