Special Angles, Reciprocal Ratios & Trig Equations (Grade 10)

The exact values for 0°, 30°, 45°, 60° and 90° without a calculator, the reciprocal ratios cosec, sec and cot, and solving simple trig equations for angles between 0° and 90°.

1The reciprocal ratios

cosec θ = 1sin θ · sec θ = 1cos θ · cot θ = 1tan θ

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The pairing is not what you would guess: sec goes with cos, and cosec goes with sin. Match the third letter: cosec ↔ sin.

These three are examined in Grade 10 only, so learn them now — they appear in Term 1 and then not again.

2The special angles

θ	0°	30°	45°	60°	90°
sin	0	12	12	32	1
cos	1	32	12	12	0
tan	0	13	1	3	undef

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Notice the sin row read backwards is the cos row. Learn one row and you have the other free.

Worked ExampleEvaluate sin30° + cos60° without a calculator

1
sin30° = 12 and cos60° = 12.
From the table.
2
12 + 12 = 1.

Worked ExampleEvaluate tan60° × cos30° without a calculator

1
tan60° = 3 and cos30° = 32.
2
3 × 32 = 32.
3 × 3 = 3.
3
= 32 or 1,5.

3Solving simple trig equations (0° to 90°)

Worked ExampleSolve 2 sinθ = 1 for θ ∈ [0° ; 90°]

1
Isolate the ratio first: sinθ = 12.
Get sinθ alone before touching the angle.
2
θ = sin⁻¹(12).
3
θ = 30°.
From the special-angle table — no calculator needed.

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2 sinθ = 1 does not mean sin(2θ) = 1. The 2 multiplies the whole ratio, so divide by it first.

Worked ExampleSolve 3 tanθ − 3 = 0 for θ ∈ [0° ; 90°]

1
3 tanθ = 3, so tanθ = 1.
2
θ = tan⁻¹(1) = 45°.

Worked ExampleSolve cos2θ = 0,5 for θ ∈ [0° ; 90°]

1
Here the 2 IS attached to the angle.
Treat 2θ as one object.
2
2θ = cos⁻¹(0,5) = 60°.
3
θ = 30°.
Divide by 2 at the very end, not the start.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Write down the value of sin30° without using a calculator.
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12
2
Evaluate sin30° + cos60° without using a calculator.
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12 + 12 = 1
3
Evaluate tan60° × cos30° without using a calculator.
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3 × 32 = 32
4
Write down the value of cosec30°.
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1sin30° = 112 = 2
5
Write down sec60°.
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1cos60° = 2
6
Write down cot45°.
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1tan45° = 1
7
Solve for θ if 2 sinθ = 1 and θ ∈ [0° ; 90°].
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sinθ = 12 → θ = 30°
8
Solve for θ if 3 tanθ − 3 = 0 and θ ∈ [0° ; 90°].
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tanθ = 1 → θ = 45°
9
Solve for θ if cos2θ = 0,5 and θ ∈ [0° ; 90°].
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2θ = 60° → θ = 30°
10
Why is tan90° undefined?
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tanθ = sinθcosθ and cos90° = 0, so it would mean dividing by zero.