Last reviewed: 7 November 2026
Updated November 2026. These Grade 9 trigonometry notes explain the sides of a right-angled triangle and the three trigonometric ratios: sine, cosine and tangent. For an acute angle θ in a right-angled triangle, sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse and tan θ = opposite ÷ adjacent. Many learners remember these as SOH CAH TOA. With these ratios you can find a missing side or a missing angle.
The worked examples and questions are original practice for Kenyan junior school learners, not an official paper. For the other Grade 9 maths topics, see our Grade 9 maths revision guide and the Grade 9 notes hub. The Pythagoras theorem notes are useful alongside this topic.
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What the Grade 9 Curriculum Covers
Trigonometry is a sub-strand under Geometry in the Kenya Institute of Curriculum Development’s Grade 9 Mathematics design. Learners identify angles and sides of right-angled triangles, identify sine, cosine and tangent ratios, read tables of trigonometric ratios for acute angles, use calculators, and apply the ratios to calculate lengths and angles. The design is listed on the KICD Grade Nine curriculum designs page. Sine and cosine rules, angles above 90° and graphs of trigonometric functions belong to later work and are left out.
Naming the Sides of a Right-Angled Triangle
- Hypotenuse: the longest side, opposite the right angle. It never changes.
- Opposite: the side facing the angle you are working with.
- Adjacent: the side next to that angle which is not the hypotenuse.
The opposite and adjacent sides swap when you switch to the other acute angle. Always mark the angle first, then label the sides.
The Three Ratios
These definitions match the ones in OpenStax Algebra and Trigonometry’s section on right triangle trigonometry.
| Ratio | Formula | Memory aid |
|---|---|---|
| sin θ | opposite ÷ hypotenuse | SOH |
| cos θ | adjacent ÷ hypotenuse | CAH |
| tan θ | opposite ÷ adjacent | TOA |
Worked example: writing the ratios
A right-angled triangle has sides 3 cm, 4 cm and 5 cm. Angle A is opposite the 3 cm side. Then sin A = 3/5 = 0.6, cos A = 4/5 = 0.8 and tan A = 3/4 = 0.75.
Reading Tables and Using a Calculator
Mathematical tables give sine, cosine and tangent for acute angles, usually to four decimal places. A calculator gives the same values; make sure it is in degree mode (often shown as D or DEG on the screen). Useful values to check your calculator: sin 30° = 0.5, cos 60° = 0.5 and tan 45° = 1. If sin 30° shows −0.988, the calculator is in radian mode.
Finding a Missing Side
Worked example: a ladder against a wall
A 5 m ladder leans against a wall and makes an angle of 60° with the ground. How high up the wall does it reach?
- The ladder is the hypotenuse (5 m). The height is opposite the 60° angle.
- Opposite and hypotenuse: use sin. sin 60° = height ÷ 5.
- Height = 5 × sin 60° = 5 × 0.8660 = 4.33 m (to 2 decimal places).
Worked example: using tan
From a point 20 m from the foot of a tree, the angle to the top of the tree is 35°. Height = 20 × tan 35° = 20 × 0.7002 = 14.0 m (to 1 decimal place). This is the kind of real-life measurement the curriculum asks learners to try outside the classroom.
When the unknown is on the bottom
If cos 40° = 12 ÷ h, then h = 12 ÷ cos 40° = 12 ÷ 0.7660 = 15.67 cm. Rearrange first, then divide; multiplying here is a common error.
Finding a Missing Angle
Work out the ratio, then use the inverse key (sin⁻¹, cos⁻¹ or tan⁻¹), or read the tables backwards. Example: opposite = 6 cm and adjacent = 8 cm. tan θ = 6 ÷ 8 = 0.75, so θ = tan⁻¹ 0.75 ≈ 36.9°.
Original Practice Questions With Worked Answers
- In a right-angled triangle, opposite = 7 cm and hypotenuse = 25 cm. Find sin θ. 7 ÷ 25 = 0.28.
- Find x if the hypotenuse is 10 cm, the angle is 30° and x is opposite the angle. x = 10 × sin 30° = 5 cm.
- A ramp rises 1.2 m over a horizontal distance of 4 m. Find its angle with the ground. tan θ = 1.2 ÷ 4 = 0.3, so θ ≈ 16.7°.
- Find the adjacent side if the hypotenuse is 15 cm and the angle is 50°. 15 × cos 50° = 15 × 0.6428 ≈ 9.64 cm.
- Which side stays the same whichever acute angle you use? The hypotenuse.
Cover the answers and attempt each question first. For timed practice, use the Grade 9 revision papers or the Grade 9 CBC Challenge. These notes are free; full revision access is available with the KSh 20 Day Pass or the KSh 500 Monthly Pass.
Frequently Asked Questions
What does SOH CAH TOA mean?
It is a memory aid: Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent.
How do I know whether to use sin, cos or tan?
Label the sides relative to the angle, note which two sides are involved (known and unknown), and pick the ratio that uses exactly those two sides.
Why does my calculator give the wrong answer for sin 30°?
It is probably in radian mode. Switch it to degree mode; sin 30° should show 0.5.
Does Grade 9 trigonometry include the sine rule?
No. Grade 9 covers sine, cosine and tangent in right-angled triangles for acute angles only.