12.3 Trigonometry: the Sine Ratio 1

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: naming the sides, the sine ratio, and finding the side opposite an angle. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Name the hypotenuse, opposite and adjacent sides for a marked angle
• Understand that sin θ = opposite ÷ hypotenuse is the same in every similar triangle
• Use a calculator to find the sine of an angle
• Use sin θ to find the side opposite an angle

Key words

• Hypotenuse (H): The longest side, opposite the right angle.
• Opposite (O): The side across from the marked angle θ.
• Adjacent (A): The side next to θ that is not the hypotenuse.
• θ (theta): A Greek letter often used for an unknown angle.
• Sine (sin): For a marked angle in a right-angled triangle, sin θ = opposite ÷ hypotenuse.

Naming the sides

Trigonometry links the angles and sides of a right-angled triangle. First mark the angle you are using, θ. The hypotenuse never changes. The opposite and adjacent sides depend on which angle you use.

Worked example: using the other angle

Triangle KLM has a right angle at L. Name the sides for the angle at M.
Hypotenuse: KM (opposite the right angle). Opposite: KL (across from M). Adjacent: LM (touches M).

Answer: H = KM, O = KL, A = LM

The sine ratio

If two right-angled triangles have the same angle θ, they are similar, so opposite ÷ hypotenuse is the same number in both. This number is called sin θ.

Key fact: sin θ = O ÷ H (remember SOH: Sine = Opposite over Hypotenuse)

• sin θ; 0° 0; 30° 0.5; 45° 0.7071…; 60° 0.8660…; 90° 1

Worked example: using a calculator

Work out sin 52° to 4 decimal places.
Check the calculator shows D or DEG (degrees). Press sin 52 = to get 0.78801…

Answer: 0.7880

Finding the opposite side

If you know the hypotenuse and the angle, the opposite side is O = H × sin θ.

Worked example: in context

A 4.5 m ladder leans against a wall at 75° to the ground. How high up the wall does it reach?
The ladder is H. The height is opposite the 75° angle.
Height = 4.5 × sin 75° = 4.3466… = 4.35 m (3 s.f.)

Answer: 4.35 m

Common mistakes

• Labelling the sides from the wrong angle, or calling the longest side “opposite”.
• Calculator in radians (R or RAD) mode: sin 30 then gives −0.988… instead of 0.5.
• Dividing instead of multiplying. If the unknown is on top (O = H × sin θ), multiply.
• Rounding sin θ before multiplying. Use the full calculator value.

Revision questions

Try these, then check your answers below each one.

Question 1. Triangle PQR has a right angle at Q. The angle θ is at P. Which side is opposite θ? (Choose from: PR, QR, PQ.)

Answer: QR

Working: QR is the side across from P. PR is the hypotenuse and PQ is adjacent.

Question 2. Use your calculator to work out sin 48°, correct to 4 decimal places.

Answer: 0.7431

Working: sin 48° = 0.74314… ≈ 0.7431.

Question 3. A right-angled triangle has hypotenuse 15 cm and an angle of 38°. Work out the side opposite the 38° angle, correct to 3 significant figures.

Answer: 9.23 cm

Working: 15 sin 38° = 9.2349… ≈ 9.23 cm.

Related lessons

• Previous lesson: 12.2 Pythagoras' Theorem 2
• Next lesson: 12.4 Trigonometry: the Sine Ratio 2
• In this chapter: 12.1 Pythagoras' Theorem 1
• In this chapter: 12.5 Trigonometry: the Cosine Ratio
• In this chapter: 12.6 Trigonometry: the Tangent Ratio

Continue in the full course

This lesson is section 12.3 (EGCMF1203) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.