Loading [a11y]/accessibility-menu.js

GCSE Tutoring Programme

Our chosen students improved 1.19 of a grade on average - 0.45 more than those who didn't have the tutoring.

Teacher-trusted tutoring

Trigonometry Formula

Here we will learn about trigonometry formulas, including the three trigonometric functions; the sine function, katex is not defined the cosine function, katex is not defined and the tangent function, katex is not defined

We will also look at the inverse trigonometric functions, the sine rule, cosine rule and the trigonometric formula for the area of a triangle.

There are also trigonometry formulas worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is a trigonometry formula?

A trigonometry formula is a formula that is used to represent relationships between the parts of a triangle including the side lengths, angles and the area.

To solve problems using trigonometric formulas, we need to be able to use the information provided to deduce which trigonometric formula we need to use.

Right-Angled Triangles

Trigonometry Formula image 1


katex is not defined

katex is not defined

katex is not defined

Non-Right Angled Triangles

Trigonometry Formula image 2


Sine Rule

katex is not defined

Cosine Rule

katex is not defined
Area of a Triangle

katex is not defined
Trigonometry Formula image 3Trigonometry Formula image 4

What is a trigonometry formula?

What is a trigonometry formula?

Trigonometry formulas for right-angled triangles

Basic trigonometry focuses on the right-angled triangle.

Trigonometry Formula image 5

The three trigonometric functions we use are sine katex is not defined cosine katex is not defined and tangent katex is not defined These three functions relate an angle to two sides of a right-angled triangle, giving us the three trigonometric ratios.

Trigonometric FunctionSine

katex is not defined
Cosine

katex is not defined
Tangent

katex is not defined
Trigonometric Ratiokatex is not definedkatex is not definedkatex is not defined

The letter katex is not defined represents a value, hence, katex is not defined is a function of the value katex is not defined

The symbol katex is not defined represents the angle.

The letter katex is not defined represents the opposite side to the angle katex is not defined.

The letter katex is not defined represents the adjacent side to the angle katex is not defined.

The letter katex is not defined represents the hypotenuse of the right-angled triangle (which is always the side of the triangle facing the right-angle). Also known as the longest side of the triangle.

Example 1

Trigonometry Formula example 1

Here,

  • the angle katex is not defined
  • the side opposite the angle katex is not defined
  • the unknown length of the hypotenuse katex is not defined

We know katex is not defined and katex is not defined and we want to find katex is not defined

Using SOHCAHTOA, SOH is the trigonometric ratio, so we need to use

katex is not defined

Example 2

Trigonometry Formula example 2

Here,

  • the angle katex is not defined
  • the hypotenuse katex is not defined
  • the unknown length of the adjacent side katex is not defined

We know katex is not defined and katex is not defined and we want to find katex is not defined

Using SOHCAHTOA, CAH is the trig ratio, so we need to use

katex is not defined

Example 3

Trigonometry Formula example 3

Here,

  • the angle katex is not defined
  • the adjacent side katex is not defined
  • the unknown length of the opposite side katex is not defined

We know katex is not defined and katex is not defined and we want to find katex is not defined

Using SOHCAHTOA, TOA is the trig ratio, so we need to use

katex is not defined

The solutions are,

Example 1: katex is not defined

Example 2: katex is not defined

Example 3: katex is not defined

Step-by-step guide: SOHCAHTOA

Inverse trigonometric functions

When we know two of the three sides of a right angled triangle and we want to find a missing angle, we need to use inverse trigonometric functions. 

The three inverse trigonometric functions are

Inverse trigonometric functionInverse Sine

katex is not defined
Inverse Cosine

katex is not defined
Inverse Tangent

katex is not defined
Inverse trig of a ratiokatex is not definedkatex is not definedkatex is not defined

Example 1

Trigonometry Formula example 1-1

Here,

  • the side opposite the angle katex is not defined
  • the length of the hypotenuse katex is not defined
  • the unknown angle is katex is not defined

We know katex is not defined and katex is not defined and we want to find katex is not defined

Using SOHCAHTOA, SOH is the trig ratio, so as we need to find the value of katex is not defined we must use the inverse trig formula katex is not defined

Example 2

Trigonometry Formula example 2-1

Here,

  • the side adjacent to the angle katex is not defined
  • the length of the hypotenuse katex is not defined
  • the unknown angle is katex is not defined

We know katex is not defined and katex is not defined and we want to find katex is not defined

Using SOHCAHTOA, CAH is the trig ratio, so as we need to find the value of katex is not defined we must use the inverse trig formula katex is not defined

Example 3

Trigonometry Formula image 6

Here,

  • the side opposite the angle katex is not defined
  • the side adjacent to the angle katex is not defined
  • the unknown angle is katex is not defined

We know katex is not defined and katex is not defined and we want to find katex is not defined

Using SOHCAHTOA, TOA is the trig ratio, so as we need to find the value of katex is not defined we must use the inverse trig formula katex is not defined

The solutions are,

Example 1: katex is not defined

Example 2: katex is not defined

Example 3: katex is not defined

Step-by-step guide: Trigonometric functions

Step-by-step guide: Trig Identities

Trigonometric formulas in non-right angled triangles

We can use trigonometry in all types of triangles including scalene, equilateral and isosceles.

Sine rule

The sine rule (sometimes called the law of sines) is the relationship between the sine of an angle, and the side opposite that angle.

More specifically the ratio of the sine of an angle and the opposing side to that angle is the same for all three angles/sides. As a formula, this is written as

katex is not defined

Where katex is not defined and katex is not defined are the three sides of a triangle, and katex is not defined and katex is not defined are the angles that face their opposing side (see the diagram below).

Trigonometry Formula image 2

Notice that the upper case letters denote the angles, and the lower case letters denote the side lengths where katex is not defined is opposite katex is not defined is opposite katex is not defined and katex is not defined is opposite katex is not defined

Trigonometry Formula image 14

The equation above shows how all three side lengths, divided by the sine of the angle facing that side, are the same value. We only need to work with a pair of letters, giving us the working formula for the law of sines.

katex is not defined

Trigonometry Formula image 8.2

We can use the sine rule to find

  • a missing side using this arrangement of the formula.

katex is not defined

By rearranging the formula to make the unknown side katex is not defined the subject, we have

katex is not defined

  • a missing angle using  this arrangement of the formula.

katex is not defined

By rearranging the formula to make the unknown angle katex is not defined the subject, we have

katex is not defined

Step-by-step guide: Sine Rule

Step-by-step guide: Changing the subject

For example,

Below is a non-right angled triangle. We can use the law of sines to find the length katex is not defined

Trigonometry Formula example image

katex is not defined

Cosine rule

The cosine rule (sometimes called the law of cosines ) is the relationship between the cosine of an angle, and all three sides of the triangle.

As a formula, this is written as

katex is not defined

Where katex is not defined and katex is not defined are the three sides of a triangle, and katex is not defined is the angle between the two sides of katex is not defined and katex is not defined (see the diagram below).

Trigonometry Formula image 6-1

Notice that the upper case katex is not defined is an angle, and the lower case letters are the side lengths where katex is not defined is opposite katex is not defined and katex is not defined and katex is not defined are the two sides that surround the angle katex is not defined

We need to use the cosine rule when we don’t know one of the four values of katex is not defined or katex is not defined

We can use the cosine rule to find

  • a missing side by using this arrangement of the formula,

katex is not defined

  • a missing angle by using this arrangement of the formula,

katex is not defined

Step-by-step guide: Cosine Rule

Step-by-step guide: Sin Cos Tan

For example,

Below is a non-right angled triangle. We can use the law of cosine to find the length katex is not defined

Trigonometry Formula image 7

katex is not defined

Area of a triangle

We can calculate the area of any triangle if we need to know two sides and the included angle between those two sides. The formula is

katex is not defined

Where katex is not defined is the angle between the two sides of katex is not defined and katex is not defined (see the diagram below).

Trigonometry Formula image 10

We can also find a missing angle knowing the area of the triangle and by rearranging the formula to find the angle katex is not defined

katex is not defined

We can find a missing side length knowing the area of the triangle and rearranging the formula to find a missing side length katex is not defined

katex is not defined

Step-by-step guide: Area of a triangle trig

For example,

We can find the area of the triangle below using the trigonometric formula for the area of any triangle.

Trigonometry Formula image 11

katex is not defined

Choosing the correct trigonometric formula

In order to determine which trigonometric formula you need to use, the flow chart below can help you decide.

Trigonometry Formula image 12

Remembering that, to find the area of any triangle, we can use the sine rule for area.

How to solve problems using trigonometry formulas

In order to solve problems using trigonometry formulas:

  1. Determine which trigonometry formula is needed to answer the question.
  2. Label each side and angle appropriately.
  3. Substitute known values into the formula.
  4. Solve for the unknown value.

Explain how to solve problems using trigonometry formulas

Explain how to solve problems using trigonometry formulas

SOHCAHTOA worksheet (includes trigonometry formulas)

SOHCAHTOA worksheet (includes trigonometry formulas)

SOHCAHTOA worksheet (includes trigonometry formulas)

Get your free trigonometry formulas worksheet of 20+ SOHCAHTOA questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE
x
SOHCAHTOA worksheet (includes trigonometry formulas)

SOHCAHTOA worksheet (includes trigonometry formulas)

SOHCAHTOA worksheet (includes trigonometry formulas)

Get your free trigonometry formulas worksheet of 20+ SOHCAHTOA questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Trigonometry formulas examples

Example 1: SOHCAHTOA (missing side)

Calculate the length of the side labelled katex is not defined Write your answer correctly to katex is not defined decimal places.

Trigonometry formulas example 1
  1. Determine which trigonometry formula is needed to answer the question.

Using the flowchart,

Trigonometry formulas example 1 step 1

This means that we need to use either of the three trig ratios of sine, cosine or tangent. We can work out which one in the next step.

2Label each side and angle appropriately.

From the angle of katex is not defined

  • the side opposite the angle katex is not defined
  • the hypotenuse katex is not defined
  • the adjacent side has no value and so is not used.

Trigonometry formulas example 1 step 2

As we know the opposite side and we want to find the hypotenuse, the trigonometric ratio that we have to use is katex is not defined

Here, we used the abbreviation katex is not defined where katex is not defined is the sine ratio katex is not defined of the opposite katex is not defined and the hypotenuse katex is not defined

3Substitute known values into the formula.

As katex is not defined and katex is not defined substituting these values into the formula

katex is not defined we have

katex is not defined

4Solve for the unknown value.

Multiplying both sides by katex is not defined gives us

katex is not defined

Dividing both sides by katex is not defined gives us

katex is not defined

This means that the length katex is not defined

Example 2: SOHCAHTOA (missing angle)

Determine the size of the missing angle, marked katex is not defined in the triangle below. Write your answer correctly to katex is not defined decimal places.

Trigonometry formulas example 2 image

Determine which trigonometry formula is needed to answer the question.

Show step

Label each side and angle appropriately.

Show step

Substitute known values into the formula.

Show step

Solve for the unknown value.

Show step

Example 3: SOHCAHTOA

Determine the missing length katex is not defined of the right-angled triangle below. Write your answer to katex is not defined decimal places.

Trigonometry formulas example 3 image

Determine which trigonometry formula is needed to answer the question.

Show step

Label each side and angle appropriately.

Show step

Substitute known values into the formula.

Show step

Solve for the unknown value.

Show step

Example 4: cosine rule

Determine the length of the side labelled katex is not defined for the triangle below. Write your answer correctly to katex is not defined decimal places.

Trigonometry formulas example 4 image

Determine which trigonometry formula is needed to answer the question.

Show step

Label each side and angle appropriately.

Show step

Substitute known values into the formula.

Show step

Solve for the unknown value.

Show step

Example 5: sine rule

Determine the size of the missing angle katex is not defined Write your answer to katex is not defined decimal places.

Trigonometry formulas example 5 image

Determine which trigonometry formula is needed to answer the question.

Show step

Label each side and angle appropriately.

Show step

Substitute known values into the formula.

Show step

Solve for the unknown value.

Show step

Example 6: area using trigonometry formulae

Calculate the area of the triangle below. Write your answer correctly to katex is not defined decimal places.

Trigonometry formulas example 6 image

Determine which trigonometry formula is needed to answer the question.

Show step

Label each side and angle appropriately.

Show step

Substitute known values into the formula.

Show step

Solve for the unknown value.

Show step

Common misconceptions

  • Labelling a triangle incorrectly

For example, this triangle has been incorrectly labelled with the side katex is not defined next to the angle rather than opposite it.

Trigonometry Formula image 13

  • Using the incorrect trigonometric function

If the triangle is incorrectly labelled, it can lead to the use of the incorrect standard or inverse trigonometric functions and formulae.

  • Rounding the decimal too early

This can lose accuracy marks. Always use as many decimal places as possible throughout the calculation, then round the final solution.

  • Using the sine rule instead of the cosine rule

In order to use the sine rule, we need to have pairs of opposite angles and sides.

  • Not using the included angle

For the cosine rule and the area of a triangle using katex is not defined the angle is included between the two sides. Using any other angle will result in an incorrect solution.

  • Using katex is not defined

If the vertical height of a triangle is not available then we cannot calculate the area by halving the base times the height.

  • Using the inverse trig function instead, inducing a mathematical error

If the inverse trig function is used instead of the standard trig function, the calculator may return a maths error as the solution does not exist.

  • Trigonometry can only be used in right-angled triangles

The use of trigonometry does not have to be limited to a right-angled triangle where all angles are acute angles. We can also use the law of cosines, the law of sines, and the area of a triangle using the sine rule on triangles that are not right-angled.

Practice trigonometry formulas questions

1. Determine the size of the missing side marked katex is not defined in the triangle below. Write your answer to katex is not defined decimal places.

Trigonometry formulas question 1

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

After labelling the triangle with katex is not defined and katex is not defined

 

  • katex is not defined
  • katex is not defined
  • katex is not defined

 

SOH CAH TOA

 

Trigonometry formulas question 1 explanation

 

katex is not defined

2. Determine the size of the missing side marked katex is not defined in the triangle below. Write your answer to katex is not defined decimal places.

Trigonometry formulas question 2

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

After labelling the triangle with katex is not defined and katex is not defined

 

  • katex is not defined
  • katex is not defined

 

SOH CAH TOA

 

Trigonometry formulas question 2 explanation

 

katex is not defined

3. Determine the size of the missing side marked katex is not defined in the triangle below. Write your answer to a suitable degree of accuracy.

 

Trigonometry formulas question 3

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

After labelling the triangle with katex is not defined and katex is not defined

 

  • katex is not defined
  • katex is not defined

 

SOH CAH TOA

 

Trigonometry formulas question 3 explanation

 

katex is not defined

4. Determine the size of the missing length marked katex is not defined in the triangle below. Write your answer to katex is not defined decimal places.

Trigonometry formulas question 4

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

Labelling the sides and angles, we have

 

Trigonometry formulas question 4 explanation

 

As we know the two side lengths of katex is not defined and katex is not defined and the included angle at katex is not defined we need to use the cosine rule (the law of cosines) to determine the length of side katex is not defined

 

katex is not defined

5. Determine the size of the missing angle marked katex is not defined in the triangle below. Write your answer to katex is not defined decimal places.

Trigonometry formulas question 5

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

Labelling the sides and angles, we have

Trigonometry formulas question 5 explanation
As we know the angle at katex is not defined the length of side katex is not defined the length of side katex is not defined and we want to determine the angle at katex is not defined we need to use the sine rule (the law of sines) to determine the length of angle katex is not defined

 

Being careful with the labelling and the correct version of the sine rule, we have

 

katex is not defined

6. Determine the area of the triangle below. Write your answer to katex is not defined decimal places.

 

Trigonometry formulas question 6

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

As we know the included angle between two side lengths, we can find the area of a triangle using sine where

 

katex is not defined and katex is not defined

 

Being careful with the labelling and the correct version of the sine rule, we have

 

katex is not defined

Trigonometry formulas GCSE questions

1. Using the triangle below, complete the table. Give your answers in exact form.

Trigonometry formulas GCSE question 1

 

Trigonometry formulas GCSE question 1 table 1

 

(3 marks)

Show answer

Trigonometry formulas GCSE question 1 table 2

 

katex is not defined correct

(1)

katex is not defined correct

(1)

katex is not defined correct

(1)

2. (a) Rearrange the equation katex is not defined to make katex is not defined the subject.

 

(b) Rearrange the equation katex is not defined to make katex is not defined the subject.

 

(c) Using your answer to part b) or otherwise, determine the missing length of the triangle katex is not defined below. Write your answer to katex is not defined decimal places.

 

Trigonometry formulas GCSE question 2

 

(7 marks)

Show answer

(a)

 

katex is not defined

(1)

katex is not defined

(1)

katex is not defined

(1)

 

(b)

 

katex is not defined

(1)

katex is not defined

(1)

 

(c)

 

katex is not defined

(1)

katex is not defined

(1)

3. (a) Show that the formula katex is not defined rearranges to make katex is not defined

 

(b) Determine the size of the missing angle katex is not defined in the triangle below. Write your answer to katex is not defined decimal places.

Trigonometry formulas GCSE question 3

 

(5 marks)

Show answer

(a)

 

katex is not defined

(1)

katex is not defined

(1)

katex is not defined

(1)

 

(b)

 

katex is not defined

(1)

katex is not defined

(1)

4. (a) Show that the formula katex is not defined rearranges to make katex is not defined

 

(b) Determine the size of the missing angle katex is not defined in the triangle below. Write your answer to katex is not defined decimal places.

 

Trigonometry formulas GCSE question 4

 

(5 marks)

Show answer

(a)

 

katex is not defined

(1)

katex is not defined

(1)

katex is not defined

(1)

 

(b)

 

katex is not defined

(1)

katex is not defined

(1)

Learning checklist

You have now learned how to:

  • Extend and formalise knowledge of ratio and proportion, including trigonometric ratios, in working with measures and geometry, and in working with proportional relations algebraically
  • Know and apply the sine rule, katex is not defined and cosine rule, katex is not defined to find unknown lengths and angles
  • Know and apply Area katex is not defined to calculate the area, sides or angles of any triangle

Beyond GCSE

Trigonometry is a large branch of mathematics. At GCSE, you barely scratch the surface of how widely trigonometry is used within mathematics and so this is developed further at A level. Other trigonometric functions you will look at at A level include

  • Secant, cosecant and cotangent reciprocal identities
    • Secant or sec katex is not defined
    • Cosecant or cosec katex is not defined
    • Cotangent or cot katex is not defined

  • Double angle formulas (the angle sum and difference formulas/sum and difference identities)
    • katex is not defined
    • katex is not defined
    • katex is not defined

  • Half angle identities
    • katex is not defined
    • katex is not defined
    • katex is not defined

  • Product identities
    • katex is not defined
    • katex is not defined
    • katex is not defined
    • katex is not defined

  • Pythagorean identities
    • katex is not defined
    • katex is not defined
    • katex is not defined

We also consider the graphs of the functions of katex is not defined and katex is not defined in greater detail in relation to when the functions are positive and negative for angles and the periodicity beyond the range of katex is not defined by using the four quadrants of an katex is not defined set of axes as well as using a different unit of measure for angles, radians. This also includes the periodicity identities for trigonometric functions in radians.

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

GCSE Benefits

Find out more about our GCSE maths tuition programme.