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Graphs of Sine, Cosine and Tangent

The three core trig functions each have a distinctive repeating graph. Knowing their shape, period and key values lets you read off solutions, spot symmetry and prepare for solving trig equations.

30 min Video by Zeeshan Zamurred Trigonometric Ratios

Edexcel AS Level Maths: 9.5 Graphs of Sine, Cosine and TangentWatch the full walkthrough before the notes below.

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What you'll be able to do

Sketch y = sin x, y = cos x and y = tan x
State the period of each function
Read key values and turning points
Use the symmetry of the graphs

Sine and cosine

Both $sin x$ and $cos x$ wave smoothly between $- 1$ and $1$ and repeat every $36 0^{\circ}$ . The cosine graph is the sine graph shifted left by $9 0^{\circ}$ — they have the same shape.

- 1 \leq sin x \leq 1, - 1 \leq cos x \leq 1, period 36 0^{\circ}

sin 0^{\circ} = 0

cos 0^{\circ} = 1

Tip — cos x = sin(x + 90°): the cosine curve is just sine slid 90° to the left.

Tangent

The tangent graph is different: it repeats every $18 0^{\circ}$ , has no maximum or minimum, and shoots off to $\pm \infty$ at vertical asymptotes where $cos x = 0$ (at $9 0^{\circ}$ , $27 0^{\circ}$ , …).

tan x = \frac{sin x}{cos x}, period 18 0^{\circ}

Asymptotes where

cos x = 0

Symmetry and key values

The graphs’ symmetry gives related angles for free: e.g. $sin (18 0^{\circ} - x) = sin x$ and $cos (- x) = cos x$ . Knowing the values at $0^{\circ}, 9 0^{\circ}, 18 0^{\circ}, 27 0^{\circ}, 36 0^{\circ}$ lets you sketch quickly and find every solution in a range.

sin x

reaches

0.5

on the way up and on the way down within one period.

Answertwo solutions (

3 0^{\circ}

and

15 0^{\circ}

)

Tip — Sketch the curve and a horizontal line at the value — the number of crossings is the number of solutions.

Formula recap

sin x, cos x : period 36 0^{\circ}, range [- 1, 1]

Sine & cosine.

tan x : period 18 0^{\circ}, asymptotes at cos x = 0

Tangent.

sin (18 0^{\circ} - x) = sin x, cos (- x) = cos x

Symmetry relations.

Common mistakes to avoid

Giving the tangent graph a maximum and minimum like sine/cosine.

tan x has no max or min; it rises without bound to its asymptotes.

Using a period of 360° for tangent.

The tangent function repeats every 180°.

Key takeaways

sin x and cos x oscillate in [−1, 1] with period 360°; cos is sin shifted 90° left.
tan x has period 180° and vertical asymptotes where cos x = 0.
Use symmetry and key values to sketch and to count solutions in a range.

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