12.4 Trigonometry: the Sine Ratio 2

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: using sin θ to find the hypotenuse and to find 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

• Rearrange sin θ = O ÷ H to find the hypotenuse
• Use the inverse sine key (sin⁻¹) to find an angle
• Recognise sin 30° = 0.5 and sin 90° = 1 without a calculator
• Solve real-life problems involving slopes and heights

Key words

• Inverse sine (sin⁻¹): Works backwards: it turns a value of sin θ into the angle θ. Usually SHIFT then sin.
• Angle of slope: The angle a ramp, road or path makes with the horizontal.
• Rearrange: Change a formula round so the unknown is on its own.
• Degrees mode: Your calculator must show D or DEG for these questions.

Finding the hypotenuse

When the unknown is the hypotenuse it is on the bottom of the fraction. Rearranging sin θ = O ÷ H gives H = O ÷ sin θ.

Check: Check: the hypotenuse must be the longest side. 12.2 > 7 ✓

Finding an angle

If you know O and H, work out O ÷ H, then use sin⁻¹ to find the angle: θ = sin⁻¹(O ÷ H).

Keys: Press SHIFT, then sin, then type (4 ÷ 9) and =. Keep the fraction in brackets.

Angles you should know

Some values come out exactly. If O is exactly half of H, the angle is 30°.

• θ; 0 0°; 0.5 30°; 1 90°

Worked example: without a calculator

A right-angled triangle has hypotenuse 12 cm. The side opposite θ is 6 cm. Find θ.
sin θ = 6 ÷ 12 = 0.5, so θ = 30°

Answer: 30°

Slopes and heights

Draw the right-angled triangle. The sloping length (ramp, ladder, string, path) is usually the hypotenuse.

Worked example: a ramp

A ramp is 5 m long and rises 0.8 m. Find the angle it makes with the ground.
sin θ = 0.8 ÷ 5 = 0.16. θ = sin⁻¹(0.16) = 9.206…° = 9.2°

Answer: 9.2°

Worked example: a kite string

A kite is 25 m above Mo’s hand. The straight string makes 48° with the horizontal. How long is the string?
O = 25, H = string. H = 25 ÷ sin 48° = 33.64… = 33.6 m

Answer: 33.6 m

Common mistakes

• Multiplying when the unknown is the hypotenuse. If x is on the bottom, divide: x = O ÷ sin θ.
• Pressing sin instead of sin⁻¹ when you want an angle.
• Working out H ÷ O instead of O ÷ H. sin θ is always less than 1.
• Rounding O ÷ H before using sin⁻¹, which can change the final angle.

Revision questions

Try these, then check your answers below each one.

Question 1. In a right-angled triangle the side opposite a 30° angle is 8 cm. Work out the hypotenuse.

Answer: 16 cm

Working: 8 ÷ sin 30° = 8 ÷ 0.5 = 16 cm.

Question 2. In a right-angled triangle the side opposite a 40° angle is 5.2 cm. Work out the hypotenuse, correct to 3 significant figures.

Answer: 8.09 cm

Working: 5.2 ÷ sin 40° = 8.0897… ≈ 8.09 cm.

Question 3. In a right-angled triangle the side opposite θ is 6 cm and the hypotenuse is 10 cm. Work out θ, correct to 1 decimal place.

Answer: 36.9 °

Working: sin⁻¹(0.6) = 36.869… ≈ 36.9°.

Related lessons

• Previous lesson: 12.3 Trigonometry: the Sine Ratio 1
• Next lesson: 12.5 Trigonometry: the Cosine Ratio
• In this chapter: 12.1 Pythagoras' Theorem 1
• In this chapter: 12.2 Pythagoras' Theorem 2
• In this chapter: 12.6 Trigonometry: the Tangent Ratio

Continue in the full course

This lesson is section 12.4 (EGCMF1204) 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.