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 = 1/(sin θ) · sec θ = 1/(cos θ) · cot θ = 1/(tan θ)
Each is undefined where the ratio underneath is 0 — cosec 0° and cot 0° do not exist, and neither does sec 90°.

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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	1/2	1/(√2)	(√3)/2	1
cos	1	(√3)/2	1/(√2)	1/2	0
tan	0	1/(√3)	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 sin 30° + cos 60° without a calculator

1
sin 30° = 1/2 and cos 60° = 1/2.
From the table.
2
1/2 + 1/2 = 1.

Worked ExampleEvaluate tan 60° × cos 30° without a calculator

1
tan 60° = √3 and cos 30° = (√3)/2.
2
√3 × (√3)/2 = 3/2.
√3 × √3 = 3.
3
= 3/2 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 = 1/2.
Get sinθ alone before touching the angle.
2
θ = sin⁻¹(1/2).
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 cos 2θ = 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 sin 30° without using a calculator.
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1/2
2
Evaluate sin 30° + cos 60° without using a calculator.
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1/2 + 1/2 = 1
3
Evaluate tan 60° × cos 30° without using a calculator.
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√3 × (√3)/2 = 3/2
4
Write down the value of cosec30°.
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1/(sin 30°) = 1/(1/2) = 2
5
Write down sec60°.
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1/(cos 60°) = 2
6
Write down cot45°.
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1/(tan 45°) = 1
7
Solve for θ if 2 sinθ = 1 and θ ∈ [0° ; 90°].
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sinθ = 1/2 → 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 cos 2θ = 0,5 and θ ∈ [0° ; 90°].
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2θ = 60° → θ = 30°
10
Why is tan 90° undefined?
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tanθ = sinθ/cosθ and cos 90° = 0, so it would mean dividing by zero.