Trigonometry

Trigonometry uses the ratios of sides in right-angled triangles to find unknown sides and angles. For Edexcel GCSE Mathematics, all students need SOHCAHTOA. Higher tier students also need exact trigonometric values, the sine rule, cosine rule and the area formula using sine.

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Key Vocabulary

Term	Meaning
Opposite	The side opposite the angle you are working with
Adjacent	The side next to the angle you are working with (not the hypotenuse)
Hypotenuse	The longest side, opposite the right angle
SOHCAHTOA	A mnemonic: Sin = Opp/Hyp, Cos = Adj/Hyp, Tan = Opp/Adj

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SOHCAHTOA

For a right-angled triangle with angle theta:

Ratio	Formula	When to use
$\sin \theta$sinθ	Opposite / Hypotenuse	You know or want Opp and Hyp
$\cos \theta$cosθ	Adjacent / Hypotenuse	You know or want Adj and Hyp
$\tan \theta$tanθ	Opposite / Adjacent	You know or want Opp and Adj

Edexcel Formula Sheet: SOHCAHTOA is NOT on the formula sheet — memorise it.

Step-by-Step Method

1. Label the sides: Opposite (O), Adjacent (A), Hypotenuse (H) relative to the angle.
2. Choose the correct ratio (which two sides are involved?).
3. Write the equation.
4. Solve for the unknown.

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Finding a Side

Worked Example 1

In a right-angled triangle, angle = 35° and hypotenuse = 12 cm. Find the opposite side.

$\sin 35°$sin35° = Opp / 12 Opp = 12 x $\sin 35°$sin35° = 12 x 0.5736... = 6.9 cm (1 d.p.)

Worked Example 2

In a right-angled triangle, angle = 50° and the adjacent side = 8 cm. Find the opposite side.

$\tan 50°$tan50° = Opp / 8 Opp = 8 x $\tan 50°$tan50° = 8 x 1.1918... = 9.5 cm (1 d.p.)

Worked Example 3

In a right-angled triangle, angle = 28° and the opposite side = 7 cm. Find the hypotenuse.

$\sin 28°$sin28° = 7 / Hyp Hyp = 7 / $\sin 28°$sin28° = 7 / 0.4695... = 14.9 cm (1 d.p.)

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Finding an Angle

Use the inverse function (sin⁻¹, cos⁻¹, tan⁻¹).

Worked Example 4

In a right-angled triangle, the opposite side is 5 cm and the hypotenuse is 9 cm. Find the angle.

$\sin \theta$sinθ = 5/9 = 0.5556... theta = $\sin^{-1}(0.5556...)$sin−1(0.5556...) = 33.7° (1 d.p.)

Worked Example 5

In a right-angled triangle, the adjacent side is 6 cm and the opposite side is 11 cm. Find the angle.

$\tan \theta$tanθ = 11/6 = 1.8333... theta = $\tan^{-1}(1.8333...)$tan−1(1.8333...) = 61.4° (1 d.p.)

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Exact Trigonometric Values [H]

You must memorise these — they are NOT on the Edexcel formula sheet:

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	$\sqrt{3}$3​/2	1/$\sqrt{3}$3​ = $\sqrt{3}$3​/3
45°	$\sqrt{2}$2​/2	$\sqrt{2}$2​/2	1
60°	$\sqrt{3}$3​/2	1/2	$\sqrt{3}$3​
90°	1	0	undefined

Worked Example 6 [H]

Find the exact length of the opposite side in a right-angled triangle with hypotenuse 10 cm and angle 30°.

$\sin 30°$sin30° = Opp / 10 Opp = 10 x 1/2 = 5 cm (exact)

Worked Example 7 [H]

A right-angled isosceles triangle has two equal sides of 1 unit. Find the exact length of the hypotenuse.

Hyp = $\sqrt{1^{2} + 1^{2}}$12+12​ = $\sqrt{2}$2​ (this confirms $\cos 45°$cos45° = 1/$\sqrt{2}$2​ = $\sqrt{2}$2​/2)

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The Sine Rule [H]

For any triangle (not just right-angled) with sides a, b, c opposite angles A, B, C:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$sinAa​=sinBb​=sinCc​

Or equivalently: $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$asinA​=bsinB​=csinC​ (use this form when finding an angle).

Edexcel Formula Sheet: The sine rule IS given on the Edexcel formula sheet.

When to Use the Sine Rule

Use when you know:

• Two angles and one side (AAS), or
• Two sides and a non-included angle (SSA — beware of the ambiguous case).

Worked Example 8 [H]

In triangle ABC: angle A = 40°, angle B = 75°, side a = 10 cm. Find side b.

a / sin A = b / sin B 10 / $\sin 40°$sin40° = b / $\sin 75°$sin75° b = 10 x $\sin 75°$sin75° / $\sin 40°$sin40° = 10 x 0.9659 / 0.6428 = 15.0 cm (1 d.p.)

Worked Example 9 [H]

In triangle PQR: PQ = 8 cm, QR = 11 cm, angle P = 85°. Find angle R.

sin R / PQ = sin P / QR sin R / 8 = $\sin 85°$sin85° / 11 sin R = 8 x $\sin 85°$sin85° / 11 = 8 x 0.9962 / 11 = 0.7245 R = $\sin^{-1}(0.7245)$sin−1(0.7245) = 46.4° (1 d.p.)

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The Cosine Rule [H]

For any triangle:

$a^2 = b^2 + c^2 - 2bc \cos A$a2=b2+c2−2bccosA (finding a side)

$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$cosA=2bcb2+c2−a2​ (finding an angle)

Edexcel Formula Sheet: The cosine rule IS given on the Edexcel formula sheet.

When to Use the Cosine Rule

Use when you know:

• Two sides and the included angle (SAS) — to find the third side.
• All three sides (SSS) — to find an angle.

Worked Example 10 [H]

In triangle ABC: b = 7 cm, c = 9 cm, angle A = 60°. Find side a.

$a^2 = 7^2 + 9^2$a2=72+92 - 2(7)(9)$\cos 60°$cos60° a² = 49 + 81 - 126 x 0.5 a² = 130 - 63 = 67 a = $\sqrt{67}$67​ = 8.2 cm (1 d.p.)

Worked Example 11 [H]

In triangle ABC: a = 5 cm, b = 7 cm, c = 9 cm. Find angle A.

cos A = (7² + 9² - 5²) / (2 x 7 x 9) cos A = (49 + 81 - 25) / 126 cos A = 105 / 126 = 5/6 = 0.8333... A = $\cos^{-1}(0.8333...)$cos−1(0.8333...) = 33.6° (1 d.p.)

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Area of a Triangle Using Sine [H]

Area $= \frac{1}{2} a b \sin C$=21​absinC

where a and b are two sides and C is the included angle.

Edexcel Formula Sheet: This formula IS given on the Edexcel formula sheet.

Worked Example 12 [H]

Find the area of a triangle with sides 8 cm and 11 cm and an included angle of 40°.

Area = 1/2 x 8 x 11 x $\sin 40°$sin40° = 44 x 0.6428 = 28.3 cm² (1 d.p.)

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Common Mistakes and Misconceptions

• Using SOHCAHTOA on non-right-angled triangles. SOHCAHTOA only works for right-angled triangles. For others, use sine/cosine rule.
• Labelling sides incorrectly. Opposite and adjacent change depending on which angle you are using.
• Calculator in wrong mode. Make sure your calculator is in DEGREES mode, not radians.
• Forgetting inverse trig when finding an angle. If $\sin \theta$sinθ = 0.5, then theta = $\sin^{-1}(0.5)$sin−1(0.5) = 30°.
• In the cosine rule, forgetting the minus sign. The formula has "- 2bc cos A" — the minus is essential.

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Edexcel Exam Tips