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 katex is not defined katex is not defined katex is not defined	Non-Right Angled Triangles Sine Rule katex is not defined Cosine Rule katex is not defined

Area of a Triangle katex is not defined

What is a trigonometry formula?

Trigonometry formulas for right-angled triangles

Basic trigonometry focuses on the right-angled triangle.

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.

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

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

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

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

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

Example 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

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

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

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).

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

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.

We can use the sine rule to find

• a missing side using this arrangement of the formula.

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

• a missing angle using  this arrangement of the formula.

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

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

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

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).

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,

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

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

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

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).

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

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

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.

Choosing the correct trigonometric formula

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

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

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.

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

Using the flowchart,

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.

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

4Solve for the unknown value.

Multiplying both sides by katex is not defined gives us

Dividing both sides by katex is not defined gives us

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.

Determine which trigonometry formula is needed to answer the question.

Show step

Using the flowchart,

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.

Label each side and angle appropriately.

Show step

From the angle of katex is not defined

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

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

However, as we need to find the missing angle, we need to use the inverse trigonometric formula katex is not defined

Here, we used the abbreviation katex is not defined where katex is not defined is the cosine ratio katex is not defined of the adjacent katex is not defined and the hypotenuse katex is not defined

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.

Determine which trigonometry formula is needed to answer the question.

Show step

Using the flowchart,

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.

Label each side and angle appropriately.

Show step

From the angle of katex is not defined

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

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

Tip: Here, we used the abbreviation katex is not defined where katex is not defined is the tangent ratio katex is not defined of the opposite katex is not defined and the adjacent katex is not defined

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.

Determine which trigonometry formula is needed to answer the question.

Show step

Using the flowchart,

For this question, we can only use the cosine rule (the law of cosines),

katex is not defined

Label each side and angle appropriately.

Show step

From the angle at katex is not defined of katex is not defined

• the side katex is not defined next to the angle katex is not defined is katex is not defined
• the side katex is not defined next to the angle katex is not defined is katex is not defined
• the side katex is not defined opposite the angle katex is not defined is katex is not defined
• the angles at katex is not defined and katex is not defined have no values and so they are not used.

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.

Determine which trigonometry formula is needed to answer the question.

Show step

Using the flowchart,

For this question, we can only use the sine rule (the law of sines),

katex is not defined

Label each side and angle appropriately.

Show step

From the angle at katex is not defined of katex is not defined

• the side katex is not defined opposite the angle katex is not defined is katex is not defined
• the side katex is not defined next to the angle katex is not defined is katex is not defined
• the angle at katex is not defined
• the angle at katex is not defined and the length of side katex is not defined have no values and so they are not used.

Example 6: area using trigonometry formulae

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

Label each side and angle appropriately.

Show step

From the angle at katex is not defined of katex is not defined

• the side katex is not defined next to the angle katex is not defined is katex is not defined
• the side katex is not defined next to the angle katex is not defined is katex is not defined
• the angles at katex is not defined and katex is not defined and the length of side katex is not defined have no values and so they are not used.

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.

• 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.

The next lessons are

• Pythagoras theorem
• Circle theorems
• Vectors

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.

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