Trigonometric graphs Worked Solutions — A-Level Maths

Recall the special angles

There are five special angles whose sine, cosine and tangent we are expected to know by heart. The value of $\sin 30^\circ$ is one of these, so we do not need a calculator.

State the exact value

From the standard results, $\sin 30^\circ$ is exactly one half. A quick check: on the sine graph the height at $30^\circ$ is halfway up to the maximum of 1.

Write the final answer

So the exact value is $\tfrac{1}{2}$. Keeping it as a fraction (not a rounded decimal) is what 'exact' means.

Recall the 45-degree values

At $45^\circ$ the sine and cosine are equal because the angle sits symmetrically between the axes. This is a fact worth memorising.

State the exact value

The exact value is $\tfrac{\sqrt2}{2}$, which is the same as $\tfrac{1}{\sqrt2}$. Both forms are correct, but rationalising the denominator gives the tidy version shown.

Write the final answer

So the exact value is $\tfrac{\sqrt2}{2}\approx 0.707$. Always leave surds in exact form unless asked to round.

Use the definition of tangent

Tangent is sine divided by cosine. We can find $\tan 60^\circ$ from the known values of $\sin 60^\circ$ and $\cos 60^\circ$.

Substitute the exact values

We know $\sin 60^\circ=\tfrac{\sqrt3}{2}$ and $\cos 60^\circ=\tfrac12$. Dividing two fractions means multiplying by the reciprocal.

Simplify

The halves cancel, leaving $\sqrt3$. So the exact value is $\sqrt3\approx 1.732$.

Convert radians to degrees

Since $\pi$ radians equals $180^\circ$, dividing by 6 gives $30^\circ$. Recognising radian versions of the special angles is essential at A-Level.

Use the known value

The sine of $30^\circ$ is one of our memorised special values. The angle in radians is exactly the same angle, so the value is unchanged.

Write the final answer

So the exact value is $\tfrac12$. It does not matter whether the angle is written in degrees or radians — it is the same point on the sine graph.

Convert radians to degrees

As $\pi=180^\circ$, dividing by 3 gives $60^\circ$. Learning to switch quickly between radians and degrees stops silly mistakes.

Use the known value

The cosine of $60^\circ$ is a standard special value equal to one half. Notice $\cos 60^\circ=\sin 30^\circ$ because the two angles are complementary.

Write the final answer

So the exact value is $\tfrac12$. Keep it as a fraction to show it is exact.