Given that t=()/(2), express in terms of t.

Write down the substitution and the t-formulae: =2t1+t^2,=1-t^21+t^2 Write down the expression to be converted: State the final expression: =2tt^2+1 Answer: 2tt^2+1

Given that t=()/(2), express in terms of t.

Write down the substitution and the t-formulae: =2t1+t^2,=1-t^21+t^2 Write down the expression to be converted: Expand the denominator: t^2+1=t^2+1 State the final expression: =1-t^2t^2+1 Answer: 1-t^2t^2+1

Given that t=()/(2), express in terms of t.

Write down the substitution and the t-formulae: =2t1+t^2,=1-t^21+t^2 Write down the expression to be converted: State the final expression: =2t1-t^2 Answer: 2t1-t^2

Given that t=()/(2), express +1 in terms of t.

Write down the substitution and the t-formulae: =2t1+t^2,=1-t^21+t^2 Write down the expression to be converted: +1 Expand the denominator: t^2+1=t^2+1 State the final expression: +1=2t^2+1 Answer: 2t^2+1

Given that t=()/(2), express 1- in terms of t.

Write down the substitution and the t-formulae: =2t1+t^2,=1-t^21+t^2 Write down the expression to be converted: 1- State the final expression: 1-=2t^2t^2+1 Answer: 2t^2t^2+1

The t-formulae Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths The t-formulae questions. See exactly how to solve problems on t-formulae, weierstrass-substitution, half-angle, expressing-in-terms-of-t.

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Further Maths70 questionsStep-by-step solutions

Question 1

2 markseasy

Given that

t=\tan\frac{\theta}{2}

, express

\sin\theta

in terms of

t

Worked solution

Write down the substitution and the t-formulae
$\sin\theta=\frac{2t}{1+t^{2}},\quad\cos\theta=\frac{1-t^{2}}{1+t^{2}}$
Every trigonometric function of $\theta$ is replaced by a rational function of $t$ .
Write down the expression to be converted
$\sin\theta$
The whole expression must be written in terms of $t$ alone.
State the final expression
$\sin\theta=\frac{2t}{t^{2}+1}$
The expression is now written entirely in terms of $t$ .

Answer

\frac{2t}{t^{2}+1}

Question 2

2 markseasy

Given that

t=\tan\frac{\theta}{2}

, express

\cos\theta

in terms of

t

Worked solution

Write down the substitution and the t-formulae
$\sin\theta=\frac{2t}{1+t^{2}},\quad\cos\theta=\frac{1-t^{2}}{1+t^{2}}$
Every trigonometric function of $\theta$ is replaced by a rational function of $t$ .
Write down the expression to be converted
$\cos\theta$
The whole expression must be written in terms of $t$ alone.
Expand the denominator
$t^{2}+1=t^{2}+1$
The denominator is a power of $1+t^{2}$ , which is never zero.
State the final expression
$\cos\theta=\frac{1-t^{2}}{t^{2}+1}$
The expression is now written entirely in terms of $t$ .

Answer

\frac{1-t^{2}}{t^{2}+1}

Question 3

2 markseasy

Given that

t=\tan\frac{\theta}{2}

, express

\tan\theta

in terms of

t

Worked solution

Write down the substitution and the t-formulae
$\sin\theta=\frac{2t}{1+t^{2}},\quad\cos\theta=\frac{1-t^{2}}{1+t^{2}}$
Every trigonometric function of $\theta$ is replaced by a rational function of $t$ .
Write down the expression to be converted
$\tan\theta$
The whole expression must be written in terms of $t$ alone.
State the final expression
$\tan\theta=\frac{2t}{1-t^{2}}$
The expression is now written entirely in terms of $t$ .

Answer

\frac{2t}{1-t^{2}}

Question 4

2 markseasy

Given that

t=\tan\frac{\theta}{2}

, express

\cos\theta+1

in terms of

t

Worked solution

Write down the substitution and the t-formulae
$\sin\theta=\frac{2t}{1+t^{2}},\quad\cos\theta=\frac{1-t^{2}}{1+t^{2}}$
Every trigonometric function of $\theta$ is replaced by a rational function of $t$ .
Write down the expression to be converted
$\cos\theta+1$
The whole expression must be written in terms of $t$ alone.
Expand the denominator
$t^{2}+1=t^{2}+1$
The denominator is a power of $1+t^{2}$ , which is never zero.
State the final expression
$\cos\theta+1=\frac{2}{t^{2}+1}$
The expression is now written entirely in terms of $t$ .

Answer

\frac{2}{t^{2}+1}

Question 5

2 markseasy

Given that

t=\tan\frac{\theta}{2}

, express

1-\cos\theta

in terms of

t

Worked solution

Write down the substitution and the t-formulae
$\sin\theta=\frac{2t}{1+t^{2}},\quad\cos\theta=\frac{1-t^{2}}{1+t^{2}}$
Every trigonometric function of $\theta$ is replaced by a rational function of $t$ .
Write down the expression to be converted
$1-\cos\theta$
The whole expression must be written in terms of $t$ alone.
State the final expression
$1-\cos\theta=\frac{2t^{2}}{t^{2}+1}$
The expression is now written entirely in terms of $t$ .

Answer

\frac{2t^{2}}{t^{2}+1}

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