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Radian measure

Radian measure

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Constant acceleration formulae: SUVAT

Vertical motion under gravity

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The normal distribution

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Finding the mean and standard deviation

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Applications to mechanics: 3D vectors

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Parametric differentiation

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Simplifying a cos x + b sin x

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Radians

Radian measure

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Areas of sectors and segments

Solving trigonometric equations

Small angle approximations

Sequences and series

Sequences revision

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Sum to infinity

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Binomial expansion II

Binomial expansion: (1+x)^n

Binomial expansion: (a+bx)^n

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Binomial expansion of compound expressions

Parametric equations

Parametric equations

Using trigonometric identities

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Points of intersection of parametric curves

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The modulus function

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Combining transformations

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Modulus inequalities

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Algebraic division

Proof II

Proof by contradiction

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Vectors I

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Magnitude and direction of a vector

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Integration I

Integrating x^n

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Differentiation I

Finding the gradient of a curve

Differentiation from first principles

Differentiating x^n

Differentiating functions with multiple terms

Gradients, tangents and normals

Increasing and decreasing functions

Second order derivatives

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Exponential modelling

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Trigonometric equations

Angles in all four quadrants

Exact trigonometric values

Trigonometric identities

Simple trigonometric equations

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Equations and identities

Trigonometry

The cosine rule

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Trig graphs: sin x, cos x, tan x

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Binomial expansion I

Pascal's triangle

Factorial notation

Binomial expansion

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Circles

Midpoints and perpendicular bisectors

Equation of a circle

Intersection of straight lines and circles

Using tangent and chord properties

Circles and triangles

Intersection of two circles

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y = mx + c

Equations of straight lines

Parallel and perpendicular lines

Length and area

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Points of intersection

Using the discriminant

Direct and inverse proportion

Polynomials

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The factor theorem

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Set notation and interval notation

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Solving disguised quadratics

Indices and surds

Laws of indices

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Surds

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Proof I

Mathematical structure and arguments

Disproof by counterexample

Proof by deduction

Proof by exhaustion

Explainer Video

Tutor: Meera

Summary

Radian measure

​​In a nutshell

So far you have measured angles in the units of degrees. Radians are an alternative unit for measuring angles, and give a much more natural description of an angle (whereas the idea of a full rotation being 360°360 \degree360° is arbitrary). It is important to know the definition of a radian, how to convert between degrees and radians and also how to use angles measured in radians in trigonometric problems.



Radians

Definition

An angle θ\thetaθ, in radians, is defined as the arc length, subtended by the angle θ\thetaθ, divided by the radius of a circle.

​θ=lr\boxed{\theta = \dfrac {l}{r}}θ=rl​​​​
  • ​θ\thetaθ is the angle in radians.​
  • ​lll is the arc length.​
  • ​rrr is the radius.​
Maths; Radians; KS5 Year 13; Radian measure

 

Consider a circle where the arc length is the circumference, so that the angle subtending the circumference gives one full rotation.


Maths; Radians; KS5 Year 13; Radian measure
Since the arc length for one full rotation is the circumference, substitute in C=2πrC=2\pi rC=2πr to give:

θ=lr=2πrr=2π\theta = \dfrac {l}{r} = \dfrac {2 \pi r}{r} = 2 \piθ=rl​=r2πr​=2π

This means that one full rotation is an angle of 2π2 \pi2π in radians. This gives the result:

360°=2π radians\boxed{360 \degree = 2 \pi \ radians}360°=2π radians​ or 180°=π radians\boxed{180 \degree = \pi \ radians}180°=π radians​​​



Converting between degrees and radians

The conversion 180°=π radians180 \degree = \pi \ radians180°=π radians can now be used to convert between degrees and radians.

→×π180180°π radians←×180π\begin{aligned}& \xrightarrow{ \times \dfrac{\pi}{180}} \\\Large 180 \degree & & \Large {\pi \ radians } \\& \xleftarrow{ \times \dfrac{180}{\pi}} \\\end {aligned}180°​×180π​​×π180​​​π radians​​​


To convert degrees to radians, ×π180\times \dfrac{\pi}{180}×180π​ and to convert radians to degrees, ×180π\times \dfrac{180}{\pi}×π180​.


Example 1

What is 90°90\degree90° in radians?


Convert from degrees into radians:

​90°=90×π180=π2 radians‾90 \degree = 90 \times \dfrac {\pi}{180} =\underline{ \dfrac {\pi}{2} \ radians}90°=90×180π​=2π​ radians​​​


Note: This is a common value that should be memorised.


Example 2

What is 11 1 radian in degrees?​


Convert from radians to degrees:

​1 radian=1×180π=57.3° (3 s.f.)‾1 \ radian = 1 \times \dfrac{180}{\pi} = \underline{57.3 \degree \ (3 \ s.f.)}1 radian=1×π180​=57.3° (3 s.f.)​​​



Trigonometric graphs

You should be able to sketch all the trig functions with the angle in radians on the xxx-axis. Note the main angles in radians: 90°=π2 rads90 \degree = \frac {\pi} {2} \ rads90°=2π​ rads, 180°=π rads180 \degree = \pi \ rads180°=π rads, 270°=3π2 rads270 \degree = \frac {3\pi} {2} \ rads270°=23π​ rads and 360°=2π rads360 \degree = 2 \pi \ rads360°=2π rads.


​y=sin⁡(x)y=\sin(x)y=sin(x)​​
​y=cos⁡(x)y= \cos(x)y=cos(x)​​
​y=tan⁡(x)y=\tan (x)y=tan(x)​​
Maths; Radians; KS5 Year 13; Radian measure
Maths; Radians; KS5 Year 13; Radian measure
Maths; Radians; KS5 Year 13; Radian measure




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Learn with Basics

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Trigonometric graphs - Higher

Unit 1

Trigonometric graphs - Higher

Trig graphs: sin x, cos x, tan x

Unit 2

Trig graphs: sin x, cos x, tan x

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Radian measure

Unit 3

Radian measure

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FAQs - Frequently Asked Questions

How big is 1 radian?

1 radian is around 57 degrees.

How do you convert from degrees to radians?

To convert from degrees to radians, multiply by pi and divide by 180.

What is a radian?

A radian is a unit for measuring angles. It is defined as the arc length subtended by the angle divided by the radius.

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