Trig Identities and Approximations

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Trig Identities and Approximations

From our first identity $textcolor{blue}{sin ^2 theta} + textcolor{limegreen}{cos ^2 theta} equiv 1$ , we have two new identities.

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Two New Identities

So, we started out in the Basic Trig Identities section by forming the identity $textcolor{blue}{sin ^2 theta} + textcolor{limegreen}{cos ^2 theta} equiv 1$ . From there, we can derive two new ones.

Identity 1:

$dfrac{textcolor{blue}{sin ^2 theta} + textcolor{limegreen}{cos ^2 theta} equiv 1}{textcolor{blue}{sin ^2 theta}}$

gives

$dfrac{textcolor{blue}{sin ^2 theta}}{textcolor{blue}{sin ^2 theta}} + dfrac{textcolor{limegreen}{cos ^2 theta}}{textcolor{blue}{sin ^2 theta}} equiv dfrac{1}{textcolor{blue}{sin ^2 theta}}$

or,

$1 + cot ^2 theta equiv cosec ^2 theta$

Identity 2:

$dfrac{textcolor{blue}{sin ^2 theta} + textcolor{limegreen}{cos ^2 theta} equiv 1}{textcolor{limegreen}{cos ^2 theta}}$

gives

$dfrac{textcolor{blue}{sin ^2 theta}}{textcolor{limegreen}{cos ^2 theta}} + dfrac{textcolor{limegreen}{cos ^2 theta}}{textcolor{limegreen}{cos ^2 theta}} equiv dfrac{1}{textcolor{limegreen}{cos ^2 theta}}$

or,

$textcolor{red}{tan ^2 theta} + 1 equiv sec ^2 theta$

Small Angle Approximations

We also have approximations for $textcolor{blue}{sin theta}$ , $textcolor{limegreen}{cos theta}$ and $textcolor{red}{tan theta}$ is small (i.e. smaller than $1$ , in radians):

$textcolor{blue}{sin theta} approx theta$

$textcolor{limegreen}{cos theta} approx 1 – dfrac{1}{2}theta ^2$

$textcolor{red}{tan theta} approx theta$

We can use these approximations to find rough values for complicated functions, in order to simplify them.

The expressions do extend to multiples of $theta$ , also, but only when the product is less than $1$ .

So, our expressions are:

$textcolor{blue}{sin ntheta} approx ntheta$

$textcolor{limegreen}{cos ntheta} approx 1 – dfrac{1}{2}(ntheta) ^2$

$textcolor{red}{tan ntheta} approx ntheta$

Example 1: Using Approximations

For small values of $theta$ , find an approximation for $dfrac{1}{2}textcolor{blue}{sin theta} + 2 textcolor{limegreen}{cos theta} – 2$ , and find any value of $theta$ where the expression is $0$ .

[3 marks]

The expression $dfrac{1}{2}textcolor{blue}{sin theta} + 2 textcolor{limegreen}{cos theta} – 2$ can be replaced by our small angle approximations to

$dfrac{1}{2}theta + 2 – theta ^2 – 2$

which can be simplified to

$dfrac{1}{2}theta – theta ^2$

When this expression is equal to zero, we have $theta ^2 – dfrac{1}{2}theta = 0$ , which has roots $theta = 0, dfrac{1}{2}$ .

Example 2: Using Identities

Solve $2sec ^2 theta = 1 + 3textcolor{red}{tan ^2 theta}$ for all values $-360° leq x leq 360°$ .

[3 marks]

First, we need to convert all trig functions to be of one form.

So,

$2 + 2textcolor{red}{tan ^2 theta} = 1 + 3textcolor{red}{tan ^2 theta}$

rearranges to

$1 = textcolor{red}{tan ^2 theta}$

so,

$textcolor{red}{tan theta} = ±1$

or,

$theta = -315°, -225°, -135°, -45°, 45°, 135°, 225°, 315°$

Trig Identities and Approximations Example Questions

Question 1: Use small angle approximations to find a value for $cos dfrac{pi}{4}$ , the form $a + bpi ^2$ .

[2 marks]

$cos theta = 1 – dfrac{1}{2}theta ^2$

So, $cos dfrac{pi}{4} = 1 – left( dfrac{1}{2} times dfrac{pi ^2}{16}right) = 1 – dfrac{pi ^2}{32}$

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Question 2: Find an approximation to $sin (3theta ) + 2cos theta$ . Use your result to find an approximate value for $sin 0.6 + 2cos 0.2$ .

What is the maximum value of $theta$ where the approximation is accurate?

[3 marks]

$sin (3theta ) + 2cos theta$

$= 3theta + 2 – theta ^2$

Using a value of $theta = 0.2$ , we have $= 3theta + 2 – theta ^2 = (3 times 0.2) + 2 – 0.2^2 = 0.6 + 2 – 0.04 = 2.56$

The maximum value of $theta$ is $dfrac{1}{3}$ .

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Question 3: Prove that $cosec ^2 theta equiv 1 + cot ^2 theta$ and $sec ^2 theta = 1 + tan ^2 theta$ .

[4 marks]

$sin ^2 theta + cos ^2 theta equiv 1$

$dfrac{sin ^2 theta}{sin ^2 theta} + dfrac{cos ^2 theta}{sin ^2 theta} equiv dfrac{1}{sin ^2 theta}$

meaning

$cosec ^2 theta equiv 1 + cot ^2 theta$

$sin ^2 theta + cos ^2 theta equiv 1$

$dfrac{sin ^2 theta}{cos ^2 theta} + dfrac{cos ^2 theta}{cos ^2 theta} equiv dfrac{1}{cos ^2 theta}$

meaning

$sec ^2 theta equiv 1 + tan ^2 theta$

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Question 4: Show that $tan ^4 theta – sec ^2 theta = 5$ can be written as $sec^4 theta – 3sec ^2 theta – 4 = 0$ and find its solutions for $0° leq theta leq 360°$ .

[5 marks]

$tan ^4 theta – sec ^2 theta = 5$

$(sec ^2 theta – 1)^2 – sec ^2 theta = 5$

$sec ^4 theta – 3sec ^2 theta – 4 = 0$

Let $x = sec ^2 theta$ .

Then we have

$x^2 – 3x – 4 = 0$

which has solutions $x = -1, 4$ .

Since $sec ^2 theta neq -1$ , we have $sec ^2 theta = 4$ , or $sec theta = ±2$ .

$sec theta = ±2$ is equivalent to $cos theta = ±dfrac{1}{2}$

So, we have

$theta = 60°, 120°, 240°, 300°$

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Additional Resources

Exam Tips Cheat Sheet

A Level

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Formula Booklet

A Level

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Specification Points Covered

E5 – Understand and use $tan{theta}=dfrac{sin{theta}}{cos{theta}}$
Understand and use $sin^2{theta}+cos^2{theta}=1$ ; $sec^2{theta}=1+tan^2{theta}$ and $cosec^2{theta}=1+cot^2{theta}$
E2 – Understand and use the standard small angle approximations of sine, cosine and tangent
$sin{theta} approx theta$ , $cos{theta} approx 1 – dfrac{theta^2}{2}$ , $tan{theta} approx theta$ where $theta$ is in radians

Trig Identities and Approximations Worksheet and Example Questions

Small Angle Approximations

Exam Questions Mark Scheme

Revise

Try it yourself

Trig Identities and Approximations

Trig Identities and Approximations

A Level Maths Predicted Papers 2025

A-A* A Level Maths Practice Papers

Two New Identities

Small Angle Approximations

Example 1: Using Approximations

Example 2: Using Identities

Trig Identities and Approximations Example Questions