Small Angle Approximations

Approximations for sin θ, cos θ, and tan θ when θ is small and measured in radians

SPEC 5.2 • TRIGONOMETRY

Learning Objectives

By the end of this lesson you will be able to:

State and apply the small angle approximations for sin θ, cos θ, and tan θ

Simplify expressions involving trigonometric functions using these approximations

Solve problems involving small angle approximations

Exam Note

Small angle approximations are frequently tested in algebraic manipulation questions. You must know these formulas – they are not in the formula booklet!

What Are Small Angle Approximations?

When an angle θ is small and measured in radians, the trigonometric functions can be approximated by simpler expressions.

Key Idea

For small values of θ (in radians):

The sine and tangent of an angle are approximately equal to the angle itself

The cosine of an angle is approximately equal to 1 minus half the angle squared

Critical

The angle θ must be in radians for these approximations to work. They do NOT work with degrees!

The Three Approximations

For small angles θ (in radians):

Sine

sin θ \approx θ

Cosine

cos θ \approx 1 - \frac{θ ^{2}}{2}

Tangent

tan θ \approx θ

Memory Aid

"Sin and Tan are Theta" — both sin θ and tan θ approximate to θ.

"Cos drops by half theta squared" — cos θ ≈ 1 − θ²/2.

Why Do They Work?

These approximations come from the Taylor series expansions of the trigonometric functions:

Taylor Series

sin θ = θ - \frac{θ ^{3}}{3 !} + \frac{θ ^{5}}{5 !} - \dots

cos θ = 1 - \frac{θ ^{2}}{2 !} + \frac{θ ^{4}}{4 !} - \dots

tan θ = θ + \frac{θ ^{3}}{3} + \frac{2 θ ^{5}}{15} + \dots

When θ is small, the higher power terms (θ³, θ⁴, ...) become negligibly small, leaving only the first terms.

Example

If θ = 0.1 rad: θ³ = 0.001, θ⁵ = 0.00001 — these are tiny!

Simplifying Expressions

Worked Example01

For small values of θ, find the approximate value of

\frac{sin 3 θ}{θ}

.

Step 1 Apply sin θ ≈ θ to sin 3θ

When θ is small, 3θ is also small, so:

sin 3 θ \approx 3 θ

Step 2 Substitute into expression

\frac{sin 3 θ}{θ} \approx \frac{3 θ}{θ} = 3

Using Cosine Approximation

Worked Example02

For small values of θ, simplify

\frac{1 - cos 2 θ}{θ ^{2}}

.

Step 1 Apply cos θ ≈ 1 − θ²/2 to cos 2θ

cos 2 θ \approx 1 - \frac{( 2 θ ) ^{2}}{2} = 1 - \frac{4 θ ^{2}}{2} = 1 - 2 θ^{2}

Step 2 Find 1 − cos 2θ

1 - cos 2 θ \approx 1 - (1 - 2 θ^{2}) = 2 θ^{2}

Step 3 Substitute into expression

\frac{1 - cos 2 θ}{θ ^{2}} \approx \frac{2 θ ^{2}}{θ ^{2}} = 2

Combined Approximations

Worked Example03

For small values of θ, show that

\frac{tan θ - sin θ}{θ ^{3}} \approx \frac{1}{2}

.

Step 1 Use more precise approximations (keeping θ³ terms)

sin θ \approx θ - \frac{θ ^{3}}{6}

and

tan θ \approx θ + \frac{θ ^{3}}{3}

Step 2 Find tan θ − sin θ

tan θ - sin θ \approx (θ + \frac{θ ^{3}}{3}) - (θ - \frac{θ ^{3}}{6})

= \frac{θ ^{3}}{3} + \frac{θ ^{3}}{6} = \frac{2 θ ^{3} + θ ^{3}}{6} = \frac{3 θ ^{3}}{6} = \frac{θ ^{3}}{2}

Step 3 Divide by θ³

\frac{tan θ - sin θ}{θ ^{3}} \approx \frac{θ ^{3} /2}{θ ^{3}} = \frac{1}{2}

✓

Ratio of Sines

Worked Example04

For small values of θ, simplify

\frac{sin 5 θ}{sin 2 θ}

.

Step 1 Apply sin θ ≈ θ to both terms

When θ is small, both 5θ and 2θ are also small, so:

sin 5 θ \approx 5 θ

and

sin 2 θ \approx 2 θ

Step 2 Substitute and simplify

\frac{sin 5 θ}{sin 2 θ} \approx \frac{5 θ}{2 θ} = \frac{5}{2}

Complex Expressions

Worked Example05

For small values of θ, simplify

\frac{1 - cos 4 θ}{sin ^{2} θ}

.

Step 1 Apply cos θ ≈ 1 − θ²/2 to cos 4θ

cos 4 θ \approx 1 - \frac{( 4 θ ) ^{2}}{2} = 1 - \frac{16 θ ^{2}}{2} = 1 - 8 θ^{2}

Step 2 Find 1 − cos 4θ and sin²θ

1 - cos 4 θ \approx 1 - (1 - 8 θ^{2}) = 8 θ^{2}

sin^{2} θ \approx θ^{2}

Step 3 Substitute and simplify

\frac{1 - cos 4 θ}{sin ^{2} θ} \approx \frac{8 θ ^{2}}{θ ^{2}} = 8

"Show That" Questions

Worked Example06

Using small angle approximations, show that for small θ:

\frac{2 sin θ + tan θ}{3 θ} \approx 1

Step 1 Apply approximations

Using sin θ ≈ θ and tan θ ≈ θ:

2 sin θ \approx 2 θ

and

tan θ \approx θ

Step 2 Substitute into numerator

2 sin θ + tan θ \approx 2 θ + θ = 3 θ

Step 3 Complete the simplification

\frac{2 sin θ + tan θ}{3 θ} \approx \frac{3 θ}{3 θ} = 1

✓

Practice

1 For small values of θ, simplify each of the following:

(a)

\frac{sin 4 θ}{tan 2 θ}

(b)

\frac{1 - cos 6 θ}{θ ^{2}}

(c)

\frac{tan 2 θ - sin 2 θ}{θ ^{3}}

(a)

\frac{sin 4 θ}{tan 2 θ} \approx \frac{4 θ}{2 θ} = 2

(b)

cos 6 θ \approx 1 - \frac{( 6 θ ) ^{2}}{2} = 1 - 18 θ^{2}

\frac{1 - cos 6 θ}{θ ^{2}} \approx \frac{18 θ ^{2}}{θ ^{2}} = 18

(c) Using tan 2θ ≈ 2θ + (2θ)³/3 and sin 2θ ≈ 2θ − (2θ)³/6:

tan 2θ − sin 2θ ≈ (8θ³/3) + (8θ³/6) = 16θ³/6 + 8θ³/6 = 24θ³/6 = 4θ³

\frac{tan 2 θ - sin 2 θ}{θ ^{3}} \approx \frac{4 θ ^{3}}{θ ^{3}} = 4

Practice

2 Using small angle approximations, show that for small θ:

\frac{3 sin 2 θ - tan 4 θ}{2 θ} \approx 1

Step 1 Apply approximations

sin 2 θ \approx 2 θ

and

tan 4 θ \approx 4 θ

Step 2 Substitute into numerator

3 sin 2 θ - tan 4 θ \approx 3 (2 θ) - 4 θ = 6 θ - 4 θ = 2 θ

Step 3 Complete the simplification

\frac{3 sin 2 θ - tan 4 θ}{2 θ} \approx \frac{2 θ}{2 θ} = 1

✓