Trigonometric Graphs: GCSE Maths Sin, Cos and Tan

Trigonometric graphs GCSE maths covers the wave-shaped curves $y = \sin x$ and $y = \cos x$, plus the repeating, asymptote-broken curve $y = \tan x$. All three graphs repeat their pattern forever, so learning the shape between $0°$ and $360°$ lets you sketch or read off any part of any of them.

What do the graphs of sin x, cos x and tan x look like?

Plotting $y = \sin x$ and $y = \cos x$ for $x$ between $0°$ and $360°$ produces identical wave shapes, just shifted along the $x$-axis. Both oscillate smoothly between $-1$ and $1$. The graph of $y = \tan x$ looks completely different: instead of a smooth wave, it rises steeply, breaks at a vertical asymptote, then starts again from very negative values.

The table below gives the exact values you should be able to recall or calculate at the key angles GCSE questions use most often.

What are the key properties of the sine and cosine graphs?

Both $y = \sin x$ and $y = \cos x$ share the same overall shape and the following properties:

• Range: every output value lies between $-1$ and $1$ inclusive, since sine and cosine can never exceed those bounds.
• Period: the pattern repeats every $360°$, meaning $\sin(x + 360°) = \sin x$ for any value of $x$.
• Symmetry: $y = \cos x$ is symmetrical about the $y$-axis, while $y = \sin x$ has rotational symmetry about the origin.

The only difference between the two graphs is a horizontal shift: $\cos x$ is the graph of $\sin x$ translated $90°$ to the left, which is why $\cos x = \sin(x + 90°)$ for every value of $x$.

Worked example: Using the graph of $y = \sin x$, find all values of $x$ between $0°$ and $360°$ for which $\sin x = 0.5$.

Sine equals $0.5$ at $x = 30°$, from the standard triangle. Because the sine graph is symmetrical about $x = 90°$, there is a second solution at $x = 180° - 30° = 150°$. Reading the graph confirms the curve crosses the horizontal line $y = 0.5$ at exactly these two points between $0°$ and $360°$, giving $x = 30°$ or $x = 150°$.

Why does the tangent graph have asymptotes?

Tangent is defined as $\tan x = \dfrac{\sin x}{\cos x}$, so wherever $\cos x = 0$, the tangent graph is undefined because you cannot divide by zero. This happens at $x = 90°$ and $x = 270°$ within one full cycle. At each of these values the graph shoots up towards positive infinity on one side and down towards negative infinity on the other, without ever touching the vertical dashed line — called an asymptote — drawn at that $x$-value.

Unlike sine and cosine, the tangent graph repeats every $180°$ rather than every $360°$, which is half the period. This shorter repeat is why the same wave pattern between each pair of asymptotes appears twice as often across a full $360°$ sweep.

How do transformations affect trigonometric graphs?

Trigonometric graphs follow the same transformation rules as any other function written in $f(x)$ form, so changes outside the trig function move the graph vertically and changes inside move it horizontally.

Worked example: Describe the graph of $y = 2\sin x$ compared with $y = \sin x$.

The "2" multiplies the whole function, so it is a vertical stretch by scale factor 2. The graph still oscillates with a period of $360°$, but its range becomes $-2$ to $2$ instead of $-1$ to $1$ — the wave gets taller without changing how often it repeats.

Worked example: Describe the graph of $y = \sin(2x)$ compared with $y = \sin x$.

The "2" is inside the function, multiplying $x$, so it is a horizontal squash by scale factor $\tfrac{1}{2}$. The graph still ranges between $-1$ and $1$, but the period halves to $180°$, meaning the wave completes two full cycles in the space one used to take.

How do you read exact trig values from the graphs?

GCSE questions often ask you to use the graph, rather than a calculator, to find a value or solve an equation. The method is the same every time:

Locate the given $x$-value on the horizontal axis, trace vertically up (or down) to the curve, then read the corresponding $y$-value from the vertical axis. To solve an equation such as $\cos x = 0$, instead draw a horizontal line at $y = 0$ and read off every $x$-value where the curve crosses it — for cosine between $0°$ and $360°$, that gives $x = 90°$ and $x = 270°$.

Frequently asked questions

What is the period of each trigonometric graph?

The sine and cosine graphs both repeat every $360°$, which is their period. The tangent graph has a shorter period of $180°$, because its pattern of rising from negative infinity to positive infinity between consecutive asymptotes repeats twice as often across a full $360°$ turn.

Why do sine and cosine graphs look identical but shifted?

Cosine and sine are the same wave shape because $\cos x = \sin(x + 90°)$ for every value of $x$. This means the cosine graph is simply the sine graph translated $90°$ to the left along the $x$-axis — the amplitude, range and period are all unchanged, only the starting position of the wave differs.

What happens to the tangent graph at 90 degrees?

At $x = 90°$, $\cos x = 0$, and since $\tan x = \sin x \div \cos x$, dividing by zero makes the function undefined. The graph does not cross $x = 90°$; instead it approaches a vertical asymptote, with the curve rising towards positive infinity just before $90°$ and reappearing from negative infinity just after it.

Do I need to memorise the whole trigonometric graph value table?

You should be able to recall or quickly derive the sine, cosine and tangent values at $0°$, $30°$, $45°$, $60°$ and $90°$, since GCSE questions frequently expect exact values without a calculator. Knowing the shape of each graph also lets you work out other angles by symmetry, rather than memorising every possible value separately.

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