GCSE Trigonometric ratios Practice Questions

Label the sides of the triangle with respect to the given angle.

The hypotenuse is the side opposite the right angle. Looking from the $30^\circ$ angle, the opposite side is the one across the triangle from it and the adjacent side is the one beside it. Here you are given the hypotenuse and asked for the opposite.

Choose the ratio that links the side you know to the side you want.

The hypotenuse and the opposite are exactly the two sides in the sine ratio (SOH), so use $\sin$. Neither of the other two ratios uses both of these sides.

State the length of the opposite side.

The opposite side is $6$ cm long.

Write down the two sides the question involves.

You have the hypotenuse and you want the adjacent side. The right ratio is the one whose fraction contains BOTH of these and nothing else.

Recall the three trigonometric ratios.

SOHCAHTOA: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.

State the ratio to use.

The cosine ratio links the hypotenuse and the adjacent side, so it is the one to use.

Label the two known sides with respect to the unknown angle.

The hypotenuse is the side opposite the right angle. Looking from the unknown angle, the opposite side is the one across the triangle from it and the adjacent side is the one beside it. Here you know the opposite and the hypotenuse.

Choose the ratio that uses the two sides you know.

The opposite and the hypotenuse are the two sides in the sine ratio (SOH), so use $\sin$. Because the ANGLE is the unknown, you will need the inverse, $\sin^{-1}$.

Write the ratio the two sides give.

The opposite goes on top and the hypotenuse underneath, so the ratio is $\frac{5}{13}$.

Undo the ratio with the inverse function.

The ratio is known and the angle is not, so apply $\sin^{-1}$ — the function that turns a ratio back into an angle.

Rule out the first wrong inverse.

$\cos^{-1}$ would be the right function only if the two sides given were the adjacent and the hypotenuse. They are not.

State the correct calculation.

The two sides given are the opposite and the hypotenuse, so the sine ratio applies, and the angle is recovered with $\sin^{-1}$.

Find the hypotenuse first.

The right angle is at $Z$, so the hypotenuse is the side opposite it, $YX$. The hypotenuse is never the opposite or the adjacent side.

Look at the triangle from the given angle.

The middle letter of $ZYX$ is $Y$, so you are standing at $Y$ and looking across the triangle. Opposite and adjacent always depend on WHICH angle you look from.

Find the opposite side.

The side across the triangle from $Y$ — the one that does not touch $Y$ at all — is $ZX$, so that is the opposite side.

Find the adjacent side.

The remaining side, $YZ$, touches $Y$ and is not the hypotenuse, so it is the adjacent side: it lies between the angle and the right angle.

Check every side has exactly one job.

The three sides take the three roles, one each. Any option that gives the hypotenuse a second job as well must be wrong.

Rule out the options that use the hypotenuse.

$YX$ is the hypotenuse, so three of the options can be discarded immediately for calling it the opposite or the adjacent side.

Rule out the option that swaps the two.

$YZ$ touches the angle at $Y$, so it cannot be the side OPPOSITE that angle. Swapping opposite and adjacent is the commonest slip here.

Note what happens at the other acute angle.

Looked at from $X$ instead, the two labels swap over. That is why you must always name the angle before you label the sides.

Recall the three trigonometric ratios.

SOHCAHTOA: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.

State the correct labelling.

From angle $ZYX$ the opposite side is $ZX$ and the adjacent side is $YZ$.

Note what the question is asking for.

Each option is a trigonometric ratio of an acute angle, so each one is just a number. Work all five out and compare them.

Check the calculator is in degrees mode.

Every angle here is in degrees. If the calculator is in radians, $\sin 30$ comes out as $-0.988\ldots$ instead of $0.5$, so test it on an angle you know before you start.

Work out the cosine of 65 degrees.

A calculator gives $\cos 65^\circ = 0.4226\ldots$.

Work out the sine of 35 degrees.

A calculator gives $\sin 35^\circ = 0.5735\ldots$, which is more than $0.4226\ldots$.

Work out the cosine of 40 degrees.

A calculator gives $\cos 40^\circ = 0.7660\ldots$, which is more than $0.4226\ldots$.

Work out the sine of 65 degrees.

A calculator gives $\sin 65^\circ = 0.9063\ldots$, which is more than $0.4226\ldots$.

Work out the tangent of 65 degrees.

A calculator gives $\tan 65^\circ = 2.1445\ldots$, which is more than $0.4226\ldots$.

Compare the five values.

Putting the five values in order shows that $0.4226\ldots$ is the smallest, so $\cos 65^\circ$ is the smallest of the five.

Note the range of sine and cosine.

For an acute angle both sine and cosine lie strictly between $0$ and $1$, so neither can ever be very large.

Note the range of tangent.

Tangent is not capped at $1$: it passes $1$ at $45^\circ$ and grows without limit as the angle approaches $90^\circ$. So a tangent can beat any sine or cosine if the angle is big enough.

Note how sine and cosine behave as the angle grows.

As an acute angle gets bigger, its sine gets bigger and its cosine gets smaller. That alone settles several of these comparisons without a calculator.

Note the connection between sine and cosine.

The sine of an angle equals the cosine of its complement, so the same value can be written in two different ways. Two options that look different can be equal.

Note where tangent overtakes sine and cosine.

Tangent passes $1$ at exactly $45^\circ$, so any tangent of an angle above $45^\circ$ beats every sine and every cosine, and any tangent below it need not.

Recall the three trigonometric ratios.

SOHCAHTOA: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.

State the smallest of the five values.

$\cos 65^\circ$ is the smallest, at $0.4226\ldots$.