1.8.2 Small angle approximations (A-level only)

What you'll learn

The standard approximations for $sin⁡θ\sin\theta$ , $cos⁡θ\cos\theta$ and $tan⁡θ\tan\theta$ when $θ\theta$ is small.
Why the angle must be measured in radians.
How to approximate numerical values and simplify trigonometric expressions.
How to solve equations that are valid for small values of $θ\theta$ .

Prerequisite: radians

An angle can be measured in degrees or radians. Radian measure is defined using a circle: an angle of $θ\theta$ radians subtends an arc whose length is $θ\theta$ times the radius.

For example, since a full turn is $2π2\pi$ radians,

radians.180^\circ=\pi\text{ radians}.

To convert degrees to radians, multiply by $π180\frac{\pi}{180}$ .

Key Idea

Radians are essential

The standard small angle approximations work only when $θ\theta$ is measured in radians. They do not work if you substitute an angle measured in degrees.

Example

Converting a small angle to radians

Express an angle of 5 degrees in radians.

Use the degree-to-radian conversion:
$θ=5×π180.\theta=5\times\frac{\pi}{180}.$
Simplify the fraction:
$θ=π36.\theta=\frac{\pi}{36}.$
As a decimal, this is approximately 0.0873 radians, which is reasonably close to zero and therefore suitable for a small angle approximation.

Common Mistake

Leaving your calculator in degree mode

If you use a calculator to check an approximation, make sure it is in radian mode. For example, $sin⁡(0.1)\sin(0.1)$ means the sine of 0.1 radians, not 0.1 degrees.

What is a small angle approximation?

An approximation is a value or expression that is close to the exact value but is usually simpler to work with.

A small angle approximation replaces a trigonometric function with a simpler algebraic expression when its angle is close to zero.

Definition

The standard approximations

For $θ\theta$ close to zero, measured in radians,

sin⁡θ≈θ,\sin\theta\approx\theta,

cos⁡θ≈1−θ22,\cos\theta\approx 1-\frac{\theta^2}{2},

and

tan⁡θ≈θ.\tan\theta\approx\theta.

The symbol $≈\approx$ means “is approximately equal to”.

The diagram shows that each exact trigonometric curve is very close to its approximation near $θ=0\theta=0$ . As the size of $θ\theta$ increases, the curves move farther apart and the approximations become less accurate.

Graphs comparing sine, cosine and tangent with their small angle approximations near zero

Why the approximations have these forms

Near zero, the graph of $y=sin⁡θy=\sin\theta$ has almost the same gradient and position as the straight line $y=θy=\theta$ . This gives $sin⁡θ≈θ\sin\theta\approx\theta$ .

Similarly, $y=tan⁡θy=\tan\theta$ is almost the same as $y=θy=\theta$ near zero, giving $tan⁡θ≈θ\tan\theta\approx\theta$ .

The cosine curve is flatter near zero, so the linear approximation $cos⁡θ≈1\cos\theta\approx 1$ is often not accurate enough. The term $−θ22-\frac{\theta^2}{2}$ accounts for the curve bending downwards.

Notice that the cosine approximation contains $θ2\theta^2$ . This also reflects the fact that cosine is an even function:

cos⁡(−θ)=cos⁡θ.\cos(-\theta)=\cos\theta.

Key Idea

Closeness to zero matters

“Small” means that the magnitude $∣θ∣|\theta|$ is close to zero. Both small positive and small negative angles can be used.

Approximating trigonometric values

To estimate a trigonometric value, identify the appropriate standard approximation and substitute the angle.

Example

Approximating sine, cosine and tangent

Estimate $sin⁡(0.08)\sin(0.08)$ , $cos⁡(0.08)\cos(0.08)$ and $tan⁡(0.08)\tan(0.08)$ .

Apply $sin⁡θ≈θ\sin\theta\approx\theta$ :
$sin⁡(0.08)≈0.08.\sin(0.08)\approx 0.08.$
Apply $cos⁡θ≈1−θ22\cos\theta\approx 1-\frac{\theta^2}{2}$ :
$cos⁡(0.08)≈1−0.0822=1−0.0032=0.9968.\begin{aligned} \cos(0.08) &\approx 1-\frac{0.08^2}{2}\\ &=1-0.0032\\ &=0.9968. \end{aligned}$
Apply $tan⁡θ≈θ\tan\theta\approx\theta$ :
$tan⁡(0.08)≈0.08.\tan(0.08)\approx 0.08.$

These estimates are close to the corresponding calculator values because 0.08 radians is close to zero.

Tip

A quick sanity check

For a small positive angle, $sin⁡θ\sin\theta$ and $tan⁡θ\tan\theta$ should be close to $θ\theta$ , while $cos⁡θ\cos\theta$ should be positive and slightly less than 1.

Angles containing a multiple of a variable

The approximation is applied to the whole angle inside the trigonometric function.

For example, if $θ\theta$ is small, then $3θ3\theta$ may also be small enough for

sin⁡(3θ)≈3θ\sin(3\theta)\approx 3\theta

and

cos⁡(3θ)≈1−(3θ)22.\cos(3\theta)\approx 1-\frac{(3\theta)^2}{2}.

Example

Approximating an expression with multiple angles

Find a small angle approximation for

2sin⁡(3θ)+cos⁡(2θ).2\sin(3\theta)+\cos(2\theta).

Replace each trigonometric function using its entire angle:
$sin⁡(3θ)≈3θ\sin(3\theta)\approx 3\theta$
and
$cos⁡(2θ)≈1−(2θ)22.\cos(2\theta)\approx 1-\frac{(2\theta)^2}{2}.$
Substitute these into the original expression:
$2sin⁡(3θ)+cos⁡(2θ)≈2(3θ)+1−(2θ)22.2\sin(3\theta)+\cos(2\theta) \approx 2(3\theta)+1-\frac{(2\theta)^2}{2}.$
Simplify:
$2sin⁡(3θ)+cos⁡(2θ)≈6θ+1−4θ22=1+6θ−2θ2.\begin{aligned} 2\sin(3\theta)+\cos(2\theta) &\approx 6\theta+1-\frac{4\theta^2}{2}\\ &=1+6\theta-2\theta^2. \end{aligned}$

Common Mistake

Forgetting to square the whole angle

In $cos⁡(3θ)≈1−(3θ)22\cos(3\theta)\approx 1-\frac{(3\theta)^2}{2}$ , the square applies to both 3 and $θ\theta$ . Therefore, $(3θ)2=9θ2(3\theta)^2=9\theta^2$ .

Simplifying ratios and expressions

Small angle approximations turn trigonometric expressions into algebraic ones. You can then expand, cancel and simplify normally.

Example

Simplifying a trigonometric ratio

Find an approximation for

1−cos⁡(2θ)θ2\frac{1-\cos(2\theta)}{\theta^2}

when $θ\theta$ is small and non-zero.

Approximate the cosine term:
$cos⁡(2θ)≈1−(2θ)22=1−2θ2.\cos(2\theta)\approx 1-\frac{(2\theta)^2}{2} =1-2\theta^2.$
Substitute this into the numerator:
$1−cos⁡(2θ)≈1−(1−2θ2)=2θ2.1-\cos(2\theta)\approx 1-(1-2\theta^2)=2\theta^2.$
Divide by $θ2\theta^2$ :
$1−cos⁡(2θ)θ2≈2θ2θ2=2.\frac{1-\cos(2\theta)}{\theta^2} \approx\frac{2\theta^2}{\theta^2}=2.$

Common Mistake

Do not substitute zero too early

The original ratio is undefined at $θ=0\theta=0$ because its denominator is zero. The approximation describes its behaviour for non-zero values of $θ\theta$ that are very close to zero.

Solving equations for small angles

If a question states that $θ\theta$ is small, you can replace the trigonometric functions with their approximations. The resulting equation is usually algebraic.

You must then check that any solution is genuinely close to zero.

Example

Solving an equation for a small angle

Given that $θ\theta$ is small, solve

3sin⁡θ=cos⁡θ.3\sin\theta=\cos\theta.

Substitute the standard approximations:
$3θ≈1−θ22.3\theta\approx 1-\frac{\theta^2}{2}.$
Rearrange into a quadratic equation:
$6θ≈2−θ2θ2+6θ−2≈0.\begin{aligned} 6\theta&\approx 2-\theta^2\\ \theta^2+6\theta-2&\approx 0. \end{aligned}$
Apply the quadratic formula:
$θ≈−6±62−4(1)(−2)2=−6±442=−3±11.\begin{aligned} \theta &\approx\frac{-6\pm\sqrt{6^2-4(1)(-2)}}{2}\\ &=\frac{-6\pm\sqrt{44}}{2}\\ &=-3\pm\sqrt{11}. \end{aligned}$
The value $−3−11-3-\sqrt{11}$ is approximately $- 6.32$ , which is not close to zero. Rejecting this leaves
$θ≈−3+11,\theta\approx -3+\sqrt{11},$
or approximately 0.317 radians.

Common Mistake

Keeping a non-small solution

An algebraic equation may produce more than one root, but the approximations were based on $θ\theta$ being close to zero. Reject any root that does not satisfy this condition.

Exam technique

In the exam

Check that every angle is in radians before applying an approximation.
Substitute the whole angle, using brackets before squaring expressions such as $2θ2\theta$ or $3θ3\theta$ .
Simplify algebraically, keep sufficient working, and reject solutions that are not close to zero.
Use your calculator in radian mode as a final accuracy check, but give the answer produced by the requested approximation.

Self review

Check yourself

What are the three standard small angle approximations?
How would you approximate $cos⁡(4θ)\cos(4\theta)$ when $θ\theta$ is small?
Why might one root of an equation need to be rejected after using small angle approximations?

What you'll learn

The standard approximations for $sin⁡θ\sin\theta$ , $cos⁡θ\cos\theta$ and $tan⁡θ\tan\theta$ when $θ\theta$ is small.
Why the angle must be measured in radians.
How to approximate numerical values and simplify trigonometric expressions.
How to solve equations that are valid for small values of $θ\theta$ .

Prerequisite: radians

An angle can be measured in degrees or radians. Radian measure is defined using a circle: an angle of $θ\theta$ radians subtends an arc whose length is $θ\theta$ times the radius.

For example, since a full turn is $2π2\pi$ radians,

radians.180^\circ=\pi\text{ radians}.

To convert degrees to radians, multiply by $π180\frac{\pi}{180}$ .

Key Idea

Radians are essential

The standard small angle approximations work only when $θ\theta$ is measured in radians. They do not work if you substitute an angle measured in degrees.

Example

Converting a small angle to radians

Express an angle of 5 degrees in radians.

Use the degree-to-radian conversion:
$θ=5×π180.\theta=5\times\frac{\pi}{180}.$
Simplify the fraction:
$θ=π36.\theta=\frac{\pi}{36}.$
As a decimal, this is approximately 0.0873 radians, which is reasonably close to zero and therefore suitable for a small angle approximation.

Common Mistake

Leaving your calculator in degree mode

If you use a calculator to check an approximation, make sure it is in radian mode. For example, $sin⁡(0.1)\sin(0.1)$ means the sine of 0.1 radians, not 0.1 degrees.

What is a small angle approximation?

An approximation is a value or expression that is close to the exact value but is usually simpler to work with.

A small angle approximation replaces a trigonometric function with a simpler algebraic expression when its angle is close to zero.

Definition

The standard approximations

For $θ\theta$ close to zero, measured in radians,

sin⁡θ≈θ,\sin\theta\approx\theta,

cos⁡θ≈1−θ22,\cos\theta\approx 1-\frac{\theta^2}{2},

and

tan⁡θ≈θ.\tan\theta\approx\theta.

The symbol $≈\approx$ means “is approximately equal to”.

Graphs comparing sine, cosine and tangent with their small angle approximations near zero

Why the approximations have these forms

Near zero, the graph of $y=sin⁡θy=\sin\theta$ has almost the same gradient and position as the straight line $y=θy=\theta$ . This gives $sin⁡θ≈θ\sin\theta\approx\theta$ .

Similarly, $y=tan⁡θy=\tan\theta$ is almost the same as $y=θy=\theta$ near zero, giving $tan⁡θ≈θ\tan\theta\approx\theta$ .

Notice that the cosine approximation contains $θ2\theta^2$ . This also reflects the fact that cosine is an even function:

cos⁡(−θ)=cos⁡θ.\cos(-\theta)=\cos\theta.

Key Idea

Closeness to zero matters

“Small” means that the magnitude $∣θ∣|\theta|$ is close to zero. Both small positive and small negative angles can be used.

Approximating trigonometric values

To estimate a trigonometric value, identify the appropriate standard approximation and substitute the angle.

Example

Approximating sine, cosine and tangent

Estimate $sin⁡(0.08)\sin(0.08)$ , $cos⁡(0.08)\cos(0.08)$ and $tan⁡(0.08)\tan(0.08)$ .

Apply $sin⁡θ≈θ\sin\theta\approx\theta$ :
$sin⁡(0.08)≈0.08.\sin(0.08)\approx 0.08.$
Apply $cos⁡θ≈1−θ22\cos\theta\approx 1-\frac{\theta^2}{2}$ :
$cos⁡(0.08)≈1−0.0822=1−0.0032=0.9968.\begin{aligned} \cos(0.08) &\approx 1-\frac{0.08^2}{2}\\ &=1-0.0032\\ &=0.9968. \end{aligned}$
Apply $tan⁡θ≈θ\tan\theta\approx\theta$ :
$tan⁡(0.08)≈0.08.\tan(0.08)\approx 0.08.$

These estimates are close to the corresponding calculator values because 0.08 radians is close to zero.

Tip

A quick sanity check

For a small positive angle, $sin⁡θ\sin\theta$ and $tan⁡θ\tan\theta$ should be close to $θ\theta$ , while $cos⁡θ\cos\theta$ should be positive and slightly less than 1.

Angles containing a multiple of a variable

The approximation is applied to the whole angle inside the trigonometric function.

For example, if $θ\theta$ is small, then $3θ3\theta$ may also be small enough for

sin⁡(3θ)≈3θ\sin(3\theta)\approx 3\theta

and

cos⁡(3θ)≈1−(3θ)22.\cos(3\theta)\approx 1-\frac{(3\theta)^2}{2}.

Example

Approximating an expression with multiple angles

Find a small angle approximation for

2sin⁡(3θ)+cos⁡(2θ).2\sin(3\theta)+\cos(2\theta).

Replace each trigonometric function using its entire angle:
$sin⁡(3θ)≈3θ\sin(3\theta)\approx 3\theta$
and
$cos⁡(2θ)≈1−(2θ)22.\cos(2\theta)\approx 1-\frac{(2\theta)^2}{2}.$
Substitute these into the original expression:
$2sin⁡(3θ)+cos⁡(2θ)≈2(3θ)+1−(2θ)22.2\sin(3\theta)+\cos(2\theta) \approx 2(3\theta)+1-\frac{(2\theta)^2}{2}.$
Simplify:
$2sin⁡(3θ)+cos⁡(2θ)≈6θ+1−4θ22=1+6θ−2θ2.\begin{aligned} 2\sin(3\theta)+\cos(2\theta) &\approx 6\theta+1-\frac{4\theta^2}{2}\\ &=1+6\theta-2\theta^2. \end{aligned}$

Common Mistake

Forgetting to square the whole angle

In $cos⁡(3θ)≈1−(3θ)22\cos(3\theta)\approx 1-\frac{(3\theta)^2}{2}$ , the square applies to both 3 and $θ\theta$ . Therefore, $(3θ)2=9θ2(3\theta)^2=9\theta^2$ .

Simplifying ratios and expressions

Small angle approximations turn trigonometric expressions into algebraic ones. You can then expand, cancel and simplify normally.

Example

Simplifying a trigonometric ratio

Find an approximation for

1−cos⁡(2θ)θ2\frac{1-\cos(2\theta)}{\theta^2}

when $θ\theta$ is small and non-zero.

Approximate the cosine term:
$cos⁡(2θ)≈1−(2θ)22=1−2θ2.\cos(2\theta)\approx 1-\frac{(2\theta)^2}{2} =1-2\theta^2.$
Substitute this into the numerator:
$1−cos⁡(2θ)≈1−(1−2θ2)=2θ2.1-\cos(2\theta)\approx 1-(1-2\theta^2)=2\theta^2.$
Divide by $θ2\theta^2$ :
$1−cos⁡(2θ)θ2≈2θ2θ2=2.\frac{1-\cos(2\theta)}{\theta^2} \approx\frac{2\theta^2}{\theta^2}=2.$

Common Mistake

Do not substitute zero too early

The original ratio is undefined at $θ=0\theta=0$ because its denominator is zero. The approximation describes its behaviour for non-zero values of $θ\theta$ that are very close to zero.

Solving equations for small angles

If a question states that $θ\theta$ is small, you can replace the trigonometric functions with their approximations. The resulting equation is usually algebraic.

You must then check that any solution is genuinely close to zero.

Example

Solving an equation for a small angle

Given that $θ\theta$ is small, solve

3sin⁡θ=cos⁡θ.3\sin\theta=\cos\theta.

Substitute the standard approximations:
$3θ≈1−θ22.3\theta\approx 1-\frac{\theta^2}{2}.$
Rearrange into a quadratic equation:
$6θ≈2−θ2θ2+6θ−2≈0.\begin{aligned} 6\theta&\approx 2-\theta^2\\ \theta^2+6\theta-2&\approx 0. \end{aligned}$
Apply the quadratic formula:
$θ≈−6±62−4(1)(−2)2=−6±442=−3±11.\begin{aligned} \theta &\approx\frac{-6\pm\sqrt{6^2-4(1)(-2)}}{2}\\ &=\frac{-6\pm\sqrt{44}}{2}\\ &=-3\pm\sqrt{11}. \end{aligned}$
The value $−3−11-3-\sqrt{11}$ is approximately $- 6.32$ , which is not close to zero. Rejecting this leaves
$θ≈−3+11,\theta\approx -3+\sqrt{11},$
or approximately 0.317 radians.

Common Mistake

Keeping a non-small solution

An algebraic equation may produce more than one root, but the approximations were based on $θ\theta$ being close to zero. Reject any root that does not satisfy this condition.

Exam technique

In the exam

Check that every angle is in radians before applying an approximation.
Substitute the whole angle, using brackets before squaring expressions such as $2θ2\theta$ or $3θ3\theta$ .
Simplify algebraically, keep sufficient working, and reject solutions that are not close to zero.
Use your calculator in radian mode as a final accuracy check, but give the answer produced by the requested approximation.

Self review

Check yourself

What are the three standard small angle approximations?
How would you approximate $cos⁡(4θ)\cos(4\theta)$ when $θ\theta$ is small?
Why might one root of an equation need to be rejected after using small angle approximations?

What you'll learn

Prerequisite: radians

What is a small angle approximation?

Why the approximations have these forms

Approximating trigonometric values

Angles containing a multiple of a variable

Simplifying ratios and expressions

Solving equations for small angles

Review 1.8.2 Small angle approximations (A-level only) by teaching Genie

1.8.2 Small angle approximations (A-level only)

What you'll learn

Prerequisite: radians

What is a small angle approximation?

Why the approximations have these forms

Approximating trigonometric values

Angles containing a multiple of a variable

Simplifying ratios and expressions

Solving equations for small angles

Review 1.8.2 Small angle approximations (A-level only) by teaching Genie

1.8.2 Small angle approximations (A-level only)

What you'll learn

Prerequisite: radians

What is a small angle approximation?

Why the approximations have these forms

Approximating trigonometric values

Angles containing a multiple of a variable

Simplifying ratios and expressions

Solving equations for small angles

Review 1.8.2 Small angle approximations (A-level only) by teaching Genie

1.8.2 Small angle approximations (A-level only) Revision Guide

1.8.2 Small angle approximations (A-level only)

What you'll learn

Prerequisite: radians

What is a small angle approximation?

Why the approximations have these forms

Approximating trigonometric values

Angles containing a multiple of a variable

Simplifying ratios and expressions

Solving equations for small angles

Review 1.8.2 Small angle approximations (A-level only) by teaching Genie

1.8.2 Small angle approximations (A-level only) Revision Guide