Graphs of Trigonometric Functions
⭐ Higher Tier Content
Trigonometric graphs show how sine, cosine and tangent change as the angle increases. Being able to sketch, understand the behaviour of, and use these graphs is important for solving equations and interpreting patterns.
Angles on Trigonometric Graphs
Angles are measured along the horizontal axis.
Angles are usually measured in degrees, increasing from left to right.
Key angles to recognise include:
\( 0^\circ \), \( 90^\circ \), \( 180^\circ \), \( 270^\circ \) and \( 360^\circ \).
The vertical axis shows the value of the trigonometric function.
The Sine Graph
The sine graph shows the values of \( \sin \theta \).
Important features of the sine graph are:
• maximum value of 1
• minimum value of -1
• passes through the origin
• repeats every \( 360^\circ \)
Key points on the sine graph include:
$$
\sin 0^\circ = 0
$$
$$
\sin 90^\circ = 1
$$
$$
\sin 180^\circ = 0
$$
$$
\sin 270^\circ = -1
$$
$$
\sin 360^\circ = 0
$$
The sine graph is smooth and wave shaped.
The Cosine Graph
The cosine graph shows the values of \( \cos \theta \).
Important features of the cosine graph are:
• maximum value of 1
• minimum value of -1
• starts at 1 when \( \theta = 0^\circ \)
• repeats every \( 360^\circ \)
Key points on the cosine graph include:
$$
\cos 0^\circ = 1
$$
$$
\cos 90^\circ = 0
$$
$$
\cos 180^\circ = -1
$$
$$
\cos 270^\circ = 0
$$
$$
\cos 360^\circ = 1
$$
The cosine graph has the same shape as the sine graph but is shifted along the horizontal axis.
The Tangent Graph
The tangent graph shows the values of \( \tan \theta \).
Important features of the tangent graph are:
• passes through the origin
• repeats every \( 180^\circ \)
• has vertical asymptotes
• values increase rapidly
Tangent is undefined where \( \cos \theta = 0 \), which occurs at:
\( 90^\circ \) and \( 270^\circ \).
These create vertical asymptotes on the graph.
The tangent graph is not wave shaped like sine and cosine.
Understanding Graph Behaviour
Trigonometric graphs are periodic, meaning they repeat regularly.
Sine and cosine:
• repeat every \( 360^\circ \)
• always stay between -1 and 1
Tangent:
• repeats every \( 180^\circ \)
• can take any value
• has breaks in the graph
Always check the scale and angle range shown on the axes
Using Trigonometric Graphs
Trigonometric graphs are used to:
• find approximate solutions to equations
• compare values of sine, cosine and tangent
• identify how many solutions an equation has
• model repeating behaviour
For example, solutions to:
$$
\sin \theta = 0.5
$$
can be found by locating where the sine graph crosses \( y = 0.5 \).
Multiple solutions may exist within one cycle.
Key Points to Remember
Sine and cosine graphs repeat every \( 360^\circ \).
Tangent graphs repeat every \( 180^\circ \).
Sine and cosine values lie between -1 and 1.
Tangent graphs have vertical asymptotes.
Trigonometric graphs are used to solve equations and model periodic behaviour.
Understanding the shape and behaviour of trigonometric graphs allows angles and values to be estimated, equations to be solved and patterns to be interpreted accurately.