Trig Equations using t-formulae (Edexcel A Level Further Maths): Revision Note

Exam code: 9FM0

Author

Mark Curtis

Last updated

5 January 2026

Trig equations using t-formulae

What are the t-formulae?

The three t-formulae state that if $t = \tan \frac{θ}{2}$ then
- $\sin θ = \frac{2 t}{1 + t^{2}}$
- $\cos θ = \frac{1 - t^{2}}{1 + t^{2}}$
- $\tan θ = \frac{2 t}{1 - t^{2}}$
They express $\sin θ$ , $\cos θ$ and $\tan θ$ in terms of one variable only, $t$
From these, you can see the reciprocals
- $cosec θ = \frac{1 + t^{2}}{2 t}$
- $\sec θ = \frac{1 + t^{2}}{1 - t^{2}}$
- $\cot θ = \frac{1 - t^{2}}{2 t}$

Examiner Tips and Tricks

You must learn the t-formulae for $\sin θ$ , $\cos θ$ and $\tan θ$ as they are not given in the formulae booklet!

How do I solve trigonometric equations using the t-formulae?

To solve trigonometric equations using t-formulae
- let $t = \tan \frac{θ}{2}$
- convert all trig functions into algebraic fractions in $t$ using
  - $\sin θ = \frac{2 t}{1 + t^{2}}$ , $\cos θ = \frac{1 - t^{2}}{1 + t^{2}}$ , $\tan θ = \frac{2 t}{1 - t^{2}}$
  - $cosec θ = \frac{1 + t^{2}}{2 t}$ , $\sec θ = \frac{1 + t^{2}}{1 - t^{2}}$ , $\cot θ = \frac{1 - t^{2}}{2 t}$
- form and solve an equation in $t$
  - This may be quadratic, cubic, quartic, ...
  - Find all the $t$ -solutions
- convert the $t$ -solutions back into $θ$ -solutions
  - using $t = \tan \frac{θ}{2}$
    - i.e. $\frac{θ}{2} = \arctan (. . .)$ so $θ = 2 \times . . .$
  - and checking the range given
You may need to adapt the t-substitution to match the equation, e.g.:
- for equations in $\sin 4 θ$ and $\cos 4 θ$ use $t = \tan 2 θ$
- for equations in $\tan \frac{x}{3}$ and $\sin \frac{x}{3}$ use $t = \tan \frac{x}{6}$

Examiner Tips and Tricks

The t-formulae can't find solutions that make the t-substitution undefined, e.g. $t = \tan \frac{θ}{2}$ would not pick up a solution at $θ = π$ .

Worked Example

Use the substitution $t = \tan \frac{x}{2}$ to solve for $0 < x < 2 π$

$2 \cos x + \sin x = 1$

giving answers to 3 significant figures where necessary.

Answer:

Write $\cos x$ in terms of its t-formula

$\cos x = \frac{1 - t^{2}}{1 + t^{2}}$

Write $\sin x$ in terms of its t-formula

$\sin x = \frac{2 t}{1 + t^{2}}$

Substitute these into the equation

$2 (\frac{1 - t^{2}}{1 + t^{2}}) + \frac{2 t}{1 + t^{2}} = 1$

Multiply both sides by $1 + t^{2}$

$2 (1 - t^{2}) + 2 t = 1 + t^{2}$

Make one side zero

$\begin{array}{rcl} 2 - 2 t^{2} + 2 t & = & 1 + t^{2} \\ 0 & = & 3 t^{2} - 2 t - 1 \end{array}$

Solve the quadratic equation, e.g. by factorisation

$\begin{array}{rcl} 0 & = & (3 t + 1) (t - 1) \\ t & = & - \frac{1}{3} or t = 1 \end{array}$

Convert each $t$ solution back into $x$ solutions using $t = \tan \frac{x}{2}$

For $\begin{array}{rcl} t & = & - \frac{1}{3} \end{array}$

$\begin{array}{rcl} \tan \frac{x}{2} & = & - \frac{1}{3} \\ \frac{x}{2} & = & \arctan (- \frac{1}{3}) + n π \\ \frac{x}{2} & = & - 0.32175 . . ., 2.81984 . . ., . . . . \\ x & = & - 0.64350 . . ., 5.63968 . . ., . . . . \end{array}$

Find $x$ solutions in the range $0 < x < 2 π$

$\begin{array}{rcl} x & = & 5.63968 . . . \end{array}$

For $\begin{array}{rcl} t & = & 1 \end{array}$

$\begin{array}{rcl} \tan \frac{x}{2} & = & 1 \\ \frac{x}{2} & = & \arctan (1) + n π \\ \frac{x}{2} & = & \frac{π}{4}, \frac{5 π}{4}, . . . \\ x & = & \frac{π}{2}, \frac{5 π}{2}, . . . \end{array}$

Find $x$ solutions in the range $0 < x < 2 π$

$\begin{array}{rcl} x & = & \frac{π}{2} \end{array}$

Present all the solutions (rounding 5.63968... to 3 s.f.)

$\begin{array}{rcl} x & = & 5.64 (3 s . f .) or x = \frac{π}{2} \end{array}$

Examiner Tips and Tricks

Other methods to solve this equation, e.g. writing in the form $R \cos (x + α)$ , score no marks if the question asks you to use t-substitutions.

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Trig Equations using t-formulae (Edexcel A Level Further Maths): Revision Note

Trig equations using t-formulae

What are the t-formulae?

How do I solve trigonometric equations using the t-formulae?

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Further Trigonometry

The t-formulae

Trig identities using t-formulae

Trig Equations using t-formulae

Modelling using t-formulae

Further Calculus

Taylor Series

Taylor Series

Limits using Series

Series Solutions to Differential Equations

Methods in Calculus

Leibnitz's Theorem

L'Hospital's Rule

The Weierstrass Substitution

Further Differential Equations

Reducing Differential Equations

Reducing First-Order Differential Equations

Reducing Second-Order Differential Equations

Coordinate Systems

Properties of Ellipses, Parabolas & Hyperbolas

Properties of Ellipses

Properties of Parabolas

Properties of Hyperbolas

Properties of Rectangular Hyperbolas

Tangent & Normals to Ellipses, Parabolas & Hyperbolas

Tangents & Normals to Ellipses

Tangents & Normals to Parabolas

Tangents & Normals to Hyperbolas

Tangents & Normals to Rectangular Hyperbolas

Loci Problems

Loci Problems

Further Vectors

Further Vectors

Vector (Cross) Product

Scalar Triple Product

Vector Geometry with Points, Lines & Planes

Shortest Distances using Vector Rules

Areas & Volumes using Vector Rules

Direction Ratios & Direction Cosines

Further Numerical Methods

Further Numerical Methods

Numerical Solutions of First-Order Differential Equations

Numerical Solutions of Second-Order Differential Equations

Simpson's Rule

Inequalities

Inequalities

Solving Inequalities involving Algebraic Fractions

Solving Inequalities involving Modulus Functions

Author: Mark Curtis