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Grade 10 Trigonometry: How to Identify the Question Type

Grade 10 trigonometry introduces sin, cos, and tan ratios in right-angled triangles, the special angle values (30°, 45°, 60°), and reading trigonometric functions from the unit circle. Identify whether a question gives you a triangle (use SOHCAHTOA) or an equation to solve.

Type 1: Using SOHCAHTOA to find a side or angle

Trigger words: "Calculate the value of x", "Find the length", "Determine the angle"

Trigger structure: A right-angled triangle is given (or can be constructed). Two measurements are known and one is unknown.

Do not confuse with: The sine rule or cosine rule, those apply to non-right-angled triangles and are Grade 11/12 content.

Method (no numbers, just the steps)

  1. Label the triangle: identify the right angle, the reference angle (θ), and the sides (opposite, adjacent, hypotenuse)
  2. Choose the correct ratio: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj
  3. Substitute the known values
  4. Solve for the unknown using algebra or the inverse trig function
Grade 10Paper 2Level 2, Routine Procedures
In the diagram, ABC, ACD and ADE are right-angled triangles. BÂE = 90° and BÂC = 30°. BC = 20 units and AD = 60 units. 4.1.1 Calculate the length of AC.

Source: DBE Mathematics Paper 2, Grade 10, November 2018, Question 4.1.1

Common mistake

Using sin when tan is needed, always identify opposite, adjacent, and hypotenuse relative to the given angle before choosing the ratio.

See the progression: same type, increasing difficulty

Easy
In a right-angled triangle, the angle is 30° and the hypotenuse is 10. Find the side opposite the 30° angle.

Practice question: not sourced from a past paper.

Medium
In right-angled triangle PQR, P̂ = 90°, QR = 15, and PR = 9. Calculate the size of Q̂.

Practice question: not sourced from a past paper.

Hard
A ladder 8 m long leans against a wall making an angle of 65° with the ground. Calculate how high up the wall the ladder reaches.

Practice question: not sourced from a past paper.

Type 2: Special angle values

Trigger words: "Without using a calculator", "Determine the exact value"

Trigger structure: The angle is 30°, 45°, or 60° (or 0° or 90°). The instruction 'without a calculator' signals that you must use known exact values.

Do not confuse with: Calculator questions: if a calculator is allowed and the angle is not special, just compute.

Method (no numbers, just the steps)

  1. Recall the special angle table: sin/cos/tan for 0°, 30°, 45°, 60°, 90°
  2. Substitute the exact values
  3. Simplify the expression: rationalize surds if needed
Grade 10Paper 2Level 2, Routine Procedures
Determine the value of the following WITHOUT using a calculator: cosec 45° / (sin 90° . tan 60°)

Source: DBE Mathematics Paper 2, Grade 10, November 2018, Question 3.3

See the progression: same type, increasing difficulty

Easy
Without a calculator, find the value of sin 30° + cos 60°.

Practice question: not sourced from a past paper.

Medium
Without a calculator, simplify: tan 45° × cos 30°

Practice question: not sourced from a past paper.

Hard
Without a calculator, determine the exact value of: (sin 45° + cos 45°)²

Practice question: not sourced from a past paper.

Type 3: Solving a basic trig equation

Trigger words: "Solve for θ", "Find the value of θ if"

Trigger structure: A trigonometric ratio is set equal to a value and you must find the angle.

Do not confuse with: Finding a side: if a number (not an angle) is unknown, use SOHCAHTOA without inverse functions.

Method (no numbers, just the steps)

  1. Isolate the trig ratio: sin θ = value, cos θ = value, or tan θ = value
  2. Use the inverse function: θ = sin⁻¹(value) etc.
  3. At Grade 10, only the principal angle in [0° ; 360°] is required, check the quadrant for additional solutions if the interval is specified
Grade 10Paper 2Level 2, Routine Procedures
Solve for x, correct to ONE decimal place, where 0° ≤ x ≤ 90°: 5 cos x + 2 = 4

Source: DBE Mathematics Paper 2, Grade 10, November 2018, Question 4.2.2

See the progression: same type, increasing difficulty

Easy
Solve for θ: cos θ = 0, where 0° ≤ θ ≤ 360°

Practice question: not sourced from a past paper.

Medium
Solve for θ: tan θ = −1, where 0° ≤ θ ≤ 360°

Practice question: not sourced from a past paper.

Hard
Solve for θ: 2 sin θ − 1 = 0, where 0° ≤ θ ≤ 360°

Practice question: not sourced from a past paper.

Words like determine and hence appear throughout this topic, see the instruction word glossary for full definitions.