Grade 10 Trigonometry

Grade 10 trigonometry: sin, cos and tan ratios, special angles, solving right-angled triangles, simple trig equations and graphs, with worked CAPS examples.

Trigonometry is the biggest topic in Grade 10 Paper 2, at about 40 of the 100 marks. It starts with the three ratios in a right-angled triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent, and their reciprocals.

You then use the ratios to solve triangles and two-dimensional problems (angles of elevation and depression), work with the special angles 0°, 30°, 45°, 60° and 90° without a calculator, extend the definitions to angles between 0° and 360° using the Cartesian plane, solve simple trig equations and sketch y = a sin θ + q, y = a cos θ + q and y = a tan θ + q.

Key concepts

  • sin, cos and tan as ratios of sides; cosec, sec and cot as reciprocals
  • Special angles 30°, 45° and 60° from the special triangles
  • Solving right-angled triangles for a side or an angle
  • Angles of elevation and depression in 2D problems
  • Trig ratios of any angle using x, y and r on the Cartesian plane
  • Graphs of y = a sin θ + q, y = a cos θ + q and y = a tan θ + q for 0° ≤ θ ≤ 360°

Worked examples

Without a calculator, find sin 30° + cos 60°.

From the special triangle, sin 30° = 1/2 and cos 60° = 1/2, so the sum is 1.

In △ABC, B̂ = 90°, AB = 5 and BC = 12. Find tan A and the length of AC.

tan A = opposite/adjacent = BC/AB = 12/5. By Pythagoras, AC = √(5² + 12²) = √169 = 13.

If 13 sin θ − 5 = 0 and θ is acute, find cos θ without a calculator.

sin θ = 5/13, so draw a right-angled triangle with opposite 5 and hypotenuse 13. The adjacent side is √(169 − 25) = 12, so cos θ = 12/13.

Common mistakes

  • Using the wrong side as 'opposite' because the angle is at a different vertex.
  • Having the calculator in radians instead of degrees.
  • Writing sin 2θ as 2 sin θ.
  • Using an acute-angle answer when the question says the angle is obtuse or reflex.

Exam tips

  • Label the sides O, A and H relative to the angle before choosing a ratio.
  • For 'without a calculator' questions, draw the special triangle or a sketch on the Cartesian plane.
  • Round angles to one decimal place and lengths to the accuracy asked.

Frequently asked questions

What are the special angles in Grade 10?

0°, 30°, 45°, 60° and 90°. You should know their sin, cos and tan values from the special triangles, without a calculator.

Are the sine and cosine rules in Grade 10?

No. The sine, cosine and area rules for non-right-angled triangles are Grade 11 work. Grade 10 uses right-angled triangles.

How much of Paper 2 is trigonometry in Grade 10?

About 40 of the 100 marks, according to the CAPS examination guidelines.

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