Trig graphs: sin, cos, tan, sec, cosec and cot

Trig graphs are the graphs of y = sin x, y = cos x and y = tan x. The sine and cosine graphs are waves that repeat every 360° and stay between −1 and 1. The tangent graph repeats every 180° and has asymptotes at 90°, 270° and so on, where it shoots off to infinity. At A-level you also meet their reciprocals: the sec, cosec and cot graphs.

The sine graph: y = sin x

Here x is the angle in degrees, from 0° to 360°, and y is sin x. The sine graph starts at the origin, rises to a maximum of 1 at 90°, crosses back through 0 at 180°, falls to a minimum of −1 at 270°, and is back at 0 at 360°. Then it repeats.

• Period: 360°. The pattern repeats every 360°.
• Amplitude: 1. The height of the wave above its centre line.
• Range: −1 ≤ sin x ≤ 1.
• Zeros: 0°, 180°, 360°, and every multiple of 180°.
• Symmetry: sin(180° − x) = sin x, so the graph is symmetrical about x = 90°.

The cosine graph: y = cos x

Here x is the angle in degrees and y is cos x. The cosine graph is the same wave as the sine graph, moved 90° to the left. It starts at its maximum of 1 at 0°, crosses zero at 90°, reaches −1 at 180°, crosses zero again at 270°, and is back to 1 at 360°.

• Period: 360°. Amplitude: 1. Range: −1 ≤ cos x ≤ 1.
• Zeros: 90°, 270°, and every 180° from there.
• Symmetry: cos(360° − x) = cos x, so the graph is symmetrical about x = 180°, and about the y-axis.

The tangent graph: y = tan x

Here x is the angle in degrees and y is tan x, drawn for y between −5 and 5. The tan graph looks completely different. It passes through 0 at 0°, 180° and 360°, and climbs more and more steeply towards 90° and 270°. At those angles tan x is undefined, because tan x = sin x ÷ cos x and cos x = 0 there. The graph has vertical asymptotes: lines it gets closer and closer to but never touches.

• Period: 180°. It repeats twice as often as sin and cos.
• Range: all real numbers. There is no amplitude.
• Asymptotes: x = 90°, x = 270°, and every 180° from there.
• Zeros: 0°, 180°, 360°.

Key points on the sin, cos and tan graphs

If you can remember these five columns, you can sketch all three graphs. In the exam, mark these points on the axes, then join them with a smooth wave (or, for tan, smooth curves that bend towards the asymptotes). The exact trig values give you the points in between, such as sin 30° = ½.

Solving trig equations with graphs

A calculator gives you one answer to an equation like sin x = 0.6. The graph shows you the others. Draw the horizontal line y = 0.6 across the graph: every crossing is a solution.

Example 1: sin x = 0.6, for 0° ≤ x ≤ 360°

The calculator gives sin⁻¹(0.6) = 36.9°. The sine graph is symmetrical about 90°, so the second solution is 180° − 36.9° = 143.1°. x = 36.9° or 143.1°.

Example 2: cos x = −0.3, for 0° ≤ x ≤ 360°

The calculator gives cos⁻¹(−0.3) = 107.5°. The cosine graph is symmetrical about 180°, so the second solution is 360° − 107.5° = 252.5°. x = 107.5° or 252.5°.

Example 3: tan x = 3, for 0° ≤ x ≤ 360°

The calculator gives tan⁻¹(3) = 71.6°. The tan graph repeats every 180°, so the next solution is 71.6° + 180° = 251.6°. x = 71.6° or 251.6°.

If α comes out negative, add 360° to bring it into range before you use the table.

Transforming trig graphs

The usual graph transformations work on trig graphs, and they change the amplitude, the period or the position of the wave.

Example 4: y = 2 sin 3x

The 2 makes the amplitude 2, so the graph goes from −2 to 2. The 3 makes the period 360° ÷ 3 = 120°, so the wave repeats three times between 0° and 360°. The first maximum is where 3x = 90°, at (30°, 2).

Here x is the angle in degrees and y = 2 sin 3x. Build waves like this with sliders, then match a target wave, in this question from the Trigonometry course.

Sec, cosec and cot graphs (A-level)

The reciprocal trig functions are sec x = 1/cos x, cosec x = 1/sin x and cot x = 1/tan x. Their graphs come straight from the sin, cos and tan graphs: wherever the original is 0, the reciprocal has an asymptote, and wherever the original is 1 or −1, so is the reciprocal.

The sec graph

Here x is the angle in degrees and y = sec x. The sec graph has asymptotes where cos x = 0, at 90° and 270°. It is made of U-shaped curves: minimum points at (0°, 1) and (360°, 1), and maximum points at (180°, −1). Its range is y ≤ −1 or y ≥ 1, and its period is 360°.

The cosec graph

Here x is the angle in degrees and y = cosec x. The cosec graph has asymptotes where sin x = 0, at 0°, 180° and 360°. It has a minimum at (90°, 1) and a maximum at (270°, −1). Its range is also y ≤ −1 or y ≥ 1, with period 360°. It is the sec graph moved 90° to the right.

The cot graph

The cot graph has asymptotes where tan x = 0 (0°, 180°, 360°) and crosses zero where tan x has its asymptotes (90°, 270°). It slopes downwards between asymptotes, repeats every 180°, and takes every real value.

Example 5: sec x = 4, for 0° ≤ x ≤ 360°

sec x = 4 means cos x = ¼. cos⁻¹(0.25) = 75.5°, and the second solution is 360° − 75.5° = 284.5°. x = 75.5° or 284.5°. On the sec graph, these are where the horizontal line y = 4 crosses the two halves of the U shapes.

At A-level the graphs are usually drawn in radians, with 360° as 2π, 180° as π and 90° as π/2. AQA A-level Maths covers the sin, cos and tan graphs, their symmetries and periodicity in E3, and the sec, cosec and cot graphs, ranges and domains in E4.

Common mistakes

• Mixing up sin and cos. sin starts at 0, cos starts at 1.
• Drawing the tan graph as a wave. It has asymptotes and no maximum or minimum.
• Only giving one solution. Use the graph to find every solution in the range.
• Getting the period of sin(bx) the wrong way round. It is 360° ÷ b, so sin 2x repeats more often, not less.
• Moving the wrong way. sin(x + 30°) moves the graph 30° to the left, not the right.
• Drawing sec and cosec between −1 and 1. Their values are never between −1 and 1.

Practice questions

1. Write down the period and the range of y = cos x.
2. Solve sin x = 0.25 for 0° ≤ x ≤ 360°.
3. Write down the amplitude and period of y = 4 cos 2x.
4. The graph of y = sin x is translated 90° to the left. Write down the equation of the new graph in terms of cos.
5. Write down the equations of the asymptotes of y = cosec x for 0° ≤ x ≤ 360°.

Answers: 1) Period 360°, range −1 ≤ y ≤ 1. 2) x = 14.5° or 165.5°. 3) Amplitude 4, period 180°. 4) y = cos x. 5) x = 0°, x = 180° and x = 360°.

Practise with a short timed paper

Trig graphs come up in GCSE Maths Higher, and the reciprocal functions in A-level Maths. The daily GCSE Maths Higher and A-level Maths: Algebra & Functions papers are short timed papers of 8 questions, with a new one every day.

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