Further Mathematics · Pearson Edexcel 9FM0 · Option FP1 · Topic 1

Further trigonometry

The t-formulae, where t = tan(θ/2), turn trigonometric expressions into algebra, which helps you solve equations and prove identities.

3 sub-topics
Taught in Year 12
32 practice questions
3 animated lessons

Practice questions (32) Lessons Worksheets All Edexcel Further Mathematics topics

What the specification says

Every sub-topic Pearson Edexcel can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

1.1The t-formulaeY12
1.2Proving identities with t-formulaeY12
1.3Solving equations with t-formulaeY12

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 32 questions in this topic, free, no account needed.

Further trigonometry

Easy

The t-formulae · Year 12 · 2 marks

t

{}= \tan\tfrac{x}{2}

{}= \tfrac{2}{3}

.

Find $\sin x$ and $\cos x$ .
(2)

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$\sin x$ ${}= \frac{2t}{1 + t^2}$ ${}= \tfrac{12}{13}$
$\cos x$ ${}= \frac{1 - t^2}{1 + t^2}$ ${}= \tfrac{5}{13}$

Total2

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Further trigonometry

Medium

Using the t-formulae · Year 12 · 5 marks

y = \dfrac{3\sin x}{3 + \cos x}

,

0 < x < \pi

.

(a)
Using the substitution $t = \tan\tfrac{x}{2}$ , show that $y = \frac{6 t}{2 t^{2} + 4}$ .
(3)
(b)
Hence find the exact value of $y$ when $x = \tfrac{\pi}{3}$ .
(2)

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Part (a)3 marks

The t-formulae. $\sin x$ ${}= \frac{2t}{1 + t^2}, \ \cos x$ ${}= \frac{1 - t^2}{1 + t^2}$
Multiply top and bottom by $1 + t^{2}$ . $y$ ${}= \frac{3\cdot 2t}{3(1 + t^2) + (1 - t^2)}$
$= \frac{6 t}{2 t^{2} + 4}$

Part (b)2 marks

$t$ ${}= \tan\tfrac{\pi}{6}$ ${}= \tfrac{1}{\sqrt{3}}$
$y = \frac{3 \sqrt{3}}{7}$

Total5

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Further trigonometry

Hard

Proving an identity with the t-formulae · Year 12 · 6 marks

(a)
Given that $t = \tan\tfrac{x}{2}$ , prove that $\frac{\sin x}{1 + \cos x} \equiv t$ .
(3)
(b)
Hence, by using $x = \tfrac{\pi}{6}$ , show that $\tan\tfrac{\pi}{12} = 2 - \sqrt{3}$ .
(3)

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Part (a)3 marks

$\sin x$ ${}= \frac{2t}{1 + t^2}, \ \cos x$ ${}= \frac{1 - t^2}{1 + t^2}$
Substitute and simplify over a common denominator. $\frac{2t}{(1 + t^2) + (1 - t^2)}$ ${}= \frac{2t}{2}$
Cancel the common factor. $= t$

Part (b)3 marks

Exact values. $\frac{\frac{1}{2}}{1 + \frac{\sqrt{3}}{2}}$ ${}= \frac{1}{2 + \sqrt{3}}$
Solve for $t$ . $t = \frac{1}{2 + \sqrt{3}}$
Rationalise the denominator. $\tan\tfrac{\pi}{12} = 2 - \sqrt{3}$

Total6

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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