Sec 2 · G1 (N(T))

Trigonometric ratios

Practise trigonometric ratios with step-by-step help. Free account.

1. Sine, cosine and tangent

A right-angled triangle with angle θ: the hypotenuse is opposite the right angle, the opposite side faces θ, and the adjacent side is next to θ.

Ratio	Formula
sine	$\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$
cosine	$\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$
tangent	$\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$

2. Finding a side

Hypotenuse 10 cm, angle 35°: the side opposite

\displaystyle \sin 35^\circ = \dfrac{x}{10}

e.g.

\displaystyle \sin 35^\circ = \dfrac{x}{10}

\displaystyle x = 10\sin 35^\circ

\displaystyle x = 5.74\text{ (3 s.f.)}

The unknown on the bottom

\displaystyle \cos 50^\circ = \dfrac{8}{x}

e.g.

\displaystyle \cos 50^\circ = \dfrac{8}{x}

\displaystyle x = \dfrac{8}{\cos 50^\circ}

\displaystyle x = 12.4\text{ (3 s.f.)}

📖 Worked example

In the diagram, $\angle KLM = 90^\circ$ . Find $x$ , correct to 3 significant figures.

Step 1 of 3

Label the sides from the 67° angle

From the 67° angle: the side

x

cm is the opposite and the side

8.5

cm is the hypotenuse.

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3. Finding an angle

Opposite 3, adjacent 5: use the inverse

\displaystyle \tan\theta = \dfrac{3}{5}

e.g.

\displaystyle \theta = \tan^{-1} \dfrac{3}{5}

\displaystyle \theta = 31.0^\circ\text{ (1 d.p.)}

📖 Worked example

In the diagram, $\angle DEF = 90^\circ$ . Find $x$ , correct to 1 decimal place.

Step 1 of 3

Label the two known sides from the angle x°

From the angle x°: the side

8

cm is the opposite and the side

12.5

cm is the hypotenuse.

Practise this →

Practise trigonometric ratios with step-by-step help

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1. Sine, cosine and tangent

Ratio	Formula
sine	$\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$
cosine	$\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$
tangent	$\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$

2. Finding a side

Hypotenuse 10 cm, angle 35°: the side opposite

\displaystyle \sin 35^\circ = \dfrac{x}{10}

e.g.

\displaystyle \sin 35^\circ = \dfrac{x}{10}

\displaystyle x = 10\sin 35^\circ

\displaystyle x = 5.74\text{ (3 s.f.)}

The unknown on the bottom

\displaystyle \cos 50^\circ = \dfrac{8}{x}

e.g.

\displaystyle \cos 50^\circ = \dfrac{8}{x}

\displaystyle x = \dfrac{8}{\cos 50^\circ}

\displaystyle x = 12.4\text{ (3 s.f.)}

📖 Worked example

In the diagram, $\angle KLM = 90^\circ$ . Find $x$ , correct to 3 significant figures.

Step 1 of 3

Label the sides from the 67° angle

From the 67° angle: the side

x

cm is the opposite and the side

8.5

cm is the hypotenuse.

Practise this →

3. Finding an angle

Opposite 3, adjacent 5: use the inverse

\displaystyle \tan\theta = \dfrac{3}{5}

e.g.

\displaystyle \theta = \tan^{-1} \dfrac{3}{5}

\displaystyle \theta = 31.0^\circ\text{ (1 d.p.)}

📖 Worked example

In the diagram, $\angle DEF = 90^\circ$ . Find $x$ , correct to 1 decimal place.

Step 1 of 3

Label the two known sides from the angle x°

From the angle x°: the side

8

cm is the opposite and the side

12.5

cm is the hypotenuse.

Practise this →