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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 youβre still stuck.
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 ![]() ![]() |
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.
Trigonometric Function | Sine
katex is not defined | Cosine
katex is not defined | Tangent
katex is not defined |
Trigonometric Ratio | katex is not defined | katex is not defined | katex 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
Here,
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 definedExample 2
Here,
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 definedExample 3
Here,
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 definedThe 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
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 function | Inverse Sine
katex is not defined | Inverse Cosine
katex is not defined | Inverse Tangent
katex is not defined |
---|---|---|---|
Inverse trig of a ratio | katex is not defined | katex is not defined | katex is not defined |
Example 1
Here,
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,
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,
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
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 definedWhere 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.
katex is not definedWe can use the sine rule to find
By rearranging the formula to make the unknown side katex is not defined the subject, we have
katex is not definedBy rearranging the formula to make the unknown angle katex is not defined the subject, we have
katex is not definedStep-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
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 definedWhere 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 donβt know one of the four values of katex is not defined or katex is not defined
We can use the cosine rule to find
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
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 definedWhere 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
katex is not definedWe 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 definedStep-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.
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.
In order to solve problems using trigonometry formulas:
Get your free trigonometry formulas worksheet of 20+ SOHCAHTOA questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREEGet your free trigonometry formulas worksheet of 20+ SOHCAHTOA questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREECalculate the length of the side labelled katex is not defined Write your answer correctly to katex is not defined decimal places.
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
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 defined4Solve for the unknown value.
Multiplying both sides by katex is not defined gives us
katex is not definedDividing both sides by katex is not defined gives us
katex is not definedThis means that the length katex is not defined
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.
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.
From the angle of katex is not defined
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
Substitute known values into the formula.
As katex is not defined and katex is not defined is unknown, substituting these values into the formula
katex is not defined we have
katex is not defined
Solve for the unknown value.
This means that the angle katex is not defined
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.
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.
From the angle of katex is not defined
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
Substitute 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
Solve for the unknown value.
Multiplying both sides by katex is not defined gives us
katex is not defined
This means that katex is not defined
The length katex is not defined
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.
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.
From the angle at katex is not defined of katex is not defined
Substitute known values into the formula.
As katex is not defined and katex is not defined substituting these into the cosine rule
katex is not defined we have
katex is not defined
Solve for the unknown value.
The length katex is not defined
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.
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.
From the angle at katex is not defined of katex is not defined
Substitute known values into the formula.
As katex is not defined and katex is not defined substituting these into the sine rule
katex is not defined we have
katex is not defined
Remember that we do not need to use all three parts of the formula for the sine rule so we only have one equals sign.
Solve for the unknown value.
Multiplying both sides by katex is not defined we have
katex is not defined
Multiplying both sides by katex is not defined we have
katex is not defined
Dividing both sides by katex is not defined we have
katex is not defined
Calculating the inverse sine of each side, we have
katex is not defined
This gives us the value katex is not defined
Calculate the area of the triangle below. Write your answer correctly to katex is not defined decimal places.
Determine which trigonometry formula is needed to answer the question.
As we are calculating the area of the triangle, we use the sine rule for area
katex is not defined
Label each side and angle appropriately.
From the angle at katex is not defined of katex is not defined
Substitute known values into the formula.
As katex is not defined and katex is not defined substituting these into the formula for the area of a triangle using the sine rule
katex is not defined we have
katex is not defined
Solve for the unknown value.
As we are finding the area, the units are in katex is not defined
This gives us the solution katex is not defined
For example, this triangle has been incorrectly labelled with the side katex is not defined next to the angle rather than opposite it.
If the triangle is incorrectly labelled, it can lead to the use of the incorrect standard or inverse trigonometric functions and formulae.
This can lose accuracy marks. Always use as many decimal places as possible throughout the calculation, then round the final solution.
In order to use the sine rule, we need to have pairs of opposite angles and sides.
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.
If the vertical height of a triangle is not available then we cannot calculate the area by halving the base times the height.
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.
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.
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.
After labelling the triangle with katex is not defined and katex is not defined
SOH CAH TOA
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.
After labelling the triangle with katex is not defined and katex is not defined
SOH CAH TOA
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.
After labelling the triangle with katex is not defined and katex is not defined
SOH CAH TOA
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.
Labelling the sides and angles, we have
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.
Labelling the sides and angles, we have
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.
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
1. Using the triangle below, complete the table. Give your answers in exact form.
(3 marks)
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.
(7 marks)
(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.
(5 marks)
(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.
(5 marks)
(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)
You have now learned how to:
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
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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