Key points about 2D and 3D trigonometry problems
Trigonometry at Higher tier may combine other parts of geometry. This could include problems that involve:
- compound shapes
- areas of triangles using trigonometry
- bearings
- angles between lines and planes in 3D shapes
Being able to identify which formula to use, sine or cosine, and when, is necessary to answer questions on this topic successfully.
Check you are confident in how to use the sine and cosine rules.
How to find the area of a triangle using the trigonometric formula
An alternative method for finding the area of a triangle is used, when two sides and the An angle between two given sides. are known.
Calculate the area of a triangle using the following A fact, rule, or principle that is expressed in terms of mathematical symbols. The plural of formula is formulae.ย :
Area of a triangle = \(\frac{1}{2}\)๐๐ sin๐ถ
In the formula ๐ and ๐ are the lengths of the two known sides and ๐ถ is the size of the angle between them.
Follow the worked example below
GCSE exam-style questions
- Calculate the area of triangle ๐๐๐ .
Give the answer to 1 decimal place.
Area = 36ยท8 cmยฒ to 1 decimal place.
- Label the sides of the triangle. Since the vertices are not called ๐ด, ๐ต and ๐ถ let vertex, ๐, with angle 130ยฐ be ๐ถ.
The choice of ๐ด and ๐ต does not matter. Let vertex ๐ be ๐ด and vertex ๐ be ๐ต.
The 12 cm side, opposite angle ๐ด, is called ๐. The 8 cm side, opposite angle ๐ต, is called ๐.
- Substitute the values of ๐, ๐ and ๐ถ into the formula to give
Area = \(\frac{1}{2}\) ร 12 ร 8 ร sin(130)
- Type \(\frac{1}{2}\) ร 12 ร 8 ร sin(130) into a scientific calculator.
Usually, the calculator will automatically open a bracket after pressing the sine button.
Remember to close the bracket after typing in the angle.
- This gives Area = 36ยท7701โฆ
Therefore, rounded to one decimal place, Area = 36ยท8 cmยฒ.
- Calculate the area of the parallelogram.
Give the answer to 1 decimal place.
Area = 63ยท6 mยฒ to 1 decimal place.
Divide the parallelogram in two by adding a diagonal. The area of the parallelogram is twice the area of the triangle.
Label the sides of the triangle.
Since the vertices are labelled, let angle 130ยฐ be ๐ถ.
The choice of ๐ด and ๐ต does not matter.
The 5 m side, opposite angle ๐ด, is called ๐.
The 13 m side, opposite angle ๐ต, is called ๐.
- Substitute the values of ๐, ๐ and ๐ถ into the formula to give
Area = \(\frac{1}{2}\) ร 12 ร 8 ร sin(130)
- Type \(\frac{1}{2}\) ร 5 ร 13 ร sin(78) into a scientific calculator.
Usually, the calculator will automatically open a bracket after pressing the sine button.
Remember to close the bracket after typing in the angle.
This gives the area of the triangle = 31ยท7897โฆ
- Find the area of the parallelogram by multiplying this by two.
31ยท7897โฆ ร 2 = 63ยท5795โฆ
Therefore, rounded to one decimal place, the area of the parallelogram = 63ยท6 mยฒ.
- The area of the triangle is 30 cmยฒ.
Calculate the size of the acute angle ๐ฅ to 1 decimal place.
๐ฅ = 59ยท0ยฐ to 1 decimal place.
Label the sides of the triangle. The 10 cm side, opposite angle ๐ด, is called ๐. The 7 cm side, opposite angle ๐ต, is called ๐.
Substitute the values of ๐, ๐, ๐ถ and the area into the formula to give 30 = \(\frac{1}{2}\) ร 10 ร 7 ร sin(๐ฅ)
\(\frac{1}{2}\) ร 7 ร 10 = 35. This simplifies to 30 = 35 ร sin(๐ฅ).
Find sin(๐ฅ) by dividing both sides by 35. This gives 30 รท 35 = sin(๐ฅ).
Work out angle, ๐ฅ, by using the inverse function of sine.
๐ฅ = sinโปยน (30/35).
- To write sinโปยน on a scientific calculator, press 'Shift' then 'sin'. Then type 30 รท 35. Remember to close the brackets.
This gives ๐ฅ = 58ยท9972โฆ
Rounded to 1 decimal place, ๐ฅ = 59ยท0ยฐ.
How to apply the sine and cosine rules
To solve problems involving non-right-angled triangles, the correct formula must be applied.
Use the sine rule on any triangle to calculate:
- a side when two angles and an opposite side are known, eg ๐, ๐ด and ๐ต
- an angle when two sides and an opposite angle are known, eg ๐, ๐ด and ๐
Use the cosine rule on any triangle to calculate:
- a side when two sides and the included angle are known, eg ๐, ๐ and ๐ถ
- an angle when all three sides are known
ย ย ย ย ย
ย ย ย ย ย
ย ย ย ย ย
- Always add any extra angles you can calculate to a diagram, using the rules of geometry.
Follow the worked example below
GCSE exam-style questions
- An oil tanker sails from ๐ท to ๐ธ and then from ๐ธ to ๐น.
๐ธ is 25 miles from ๐ท, on a bearing of 050ยฐ.
๐น is 32 miles from ๐ธ, on a bearing of 105ยฐ.
Work out the direct distance from ๐ท to ๐น.
๐ท๐น = 50ยท7 miles, to one decimal place.
- First calculate the other angles using geometry. The two north lines are parallel.
Co-interior angles add up to 180ยฐ. The angle to the left of ๐ธ = 180 โ 50 = 130ยฐ.
Use angles at a point to calculate angle ๐ท๐ธ๐น.
Angle ๐ท๐ธ๐น = 360 โ 105 โ 130 = 125ยฐ.
In the triangle two sides and the included angle are known. The cosine rule is used to work out the missing side.
- Label the sides of the triangle.
Here the vertices are not labelled using ๐ด, ๐ต and ๐ถ, so rename the vertex with angle 125ยฐ as ๐ด.
The choice of ๐ต and ๐ does not matter. Let ๐ท be ๐ต and ๐น be ๐ถ.
The 25-mile side, opposite angle ๐ถ, is called ๐.
The 32-mile side, opposite angle ๐ต, is called ๐.
The side ๐ต๐, opposite angle ๐ด, is called ๐.
- Substitute the values of ๐ด, ๐, ๐ and ๐ into the formula to give ๐ท๐นยฒ = 32ยฒ + 25ยฒ โ (2 ร 32 ร 25) cos(125).
32ยฒ = 1024 , 25ยฒ = 625 and 2 ร 32 ร 25 = 1600, so this simplifies to ๐ท๐นยฒ = 1024 + 625 โ 1600cos(125).
- Type 1024 + 625 โ 1600cos(125) into a scientific calculator.
Usually, the calculator will automatically open a bracket after pressing the cosine button.
Remember to close the bracket after typing in the angle. This gives ๐ท๐นยฒ = 2566ยท7222โฆ
It is important not to round the number at this stage.
- Find ๐ท๐น by calculating the square root of 2566ยท7222โฆ
Type the square root button followed by the โAnsโ button into a scientific calculator.
This gives the answer of ๐ท๐น = 50ยท6628โฆ. Therefore, rounded to one decimal place, ๐ท๐น = 50ยท7 miles.
- Quadrilateral ๐๐๐๐ is formed by combining two triangles.
Work out the length of ๐๐. Give the answer to one decimal place.
๐๐ = 13ยท4 cm, to one decimal place.
Split the quadrilateral into two separate triangles and apply non-right angled trigonometry.
Step 1. Calculate the length of ๐๐ using the sine rule.
- Label the sides of the triangle. Since the vertices are not called ๐ด, ๐ต and ๐ถ let vertex ๐ be ๐ด, ๐ be ๐ต and ๐ be ๐ถโ.
The 12 cm side, opposite angle ๐ต, is called ๐. The side ๐๐ or ๐ด๐ต, opposite angle ๐ถ, is called ๐, and the unknown side, opposite angle ๐ด, is called ๐.
Since neither side ๐, or angle ๐ด is known this is the portion of the sine rule formula that will not be used.
Substitute the values of ๐ต, ๐ถ, ๐ and ๐ into the formula to give \(\frac{12}{sin(66)}\) = \(\frac{๐ด๐ต}{sin(97)}\).
Rearrange the equation to make ๐ด๐ต the subject. Find the value of ๐ด๐ต by multiplying both sides of the equation by sin(97). This gives \(\frac{12sin(97)}{sin(66)}\) = ๐ด๐ต.
Type 12sin(97) รท sin(66) into a scientific calculator. This gives ๐ด๐ต = 13ยท0377 โฆ It is important not to round the number at this stage.
Step 2. Calculate the length of ๐๐ using the cosine rule.
- Label the sides of the triangle. Keep ๐ as ๐ด, ๐ as ๐ต and let ๐ be ๐ถโ.
The 9 cm side, opposite angle ๐ต, is called ๐. The side, opposite angle ๐ถโ, is called ๐, and is the answer that was calculated in Step 1.
The side labelled ๐๐, opposite angle ๐ด, is called ๐.
- Substitute the values of ๐ด, ๐, ๐ and ๐ into the formula to give ๐๐ยฒ = 92 + ๐๐ยฒ - (2 ร 10 ร ๐๐) cos(72).
92 = 81 and 2 ร 9 ร ๐๐ = 18๐๐, so this simplifies to ๐ยฒ = 81+ ๐๐ยฒ โ 18๐๐cos(72).
- Type 81 + ๐๐ยฒ โ 18๐๐cos(72) into a scientific calculator. Use the โAnsโ button each time you need ๐๐.
This gives ๐๐ยฒ = 178ยท4642โฆ
It is again important not to round the number at this stage.
- Find ๐๐, by calculating the square root of 178ยท4642โฆ
Type the square root button followed by the โAnsโ button into a scientific calculator.
This gives the answer of ๐๐ = 13ยท3589โฆ.
Therefore, rounded to one decimal place, ๐๐ = 13ยท4 cm.
How to find the angle between a line and a plane
The angle between a line and a A two-dimensional surface. is the smallest angle between the line and its A correspondence between points of one figure and a second surface. onto the plane.
The projection is a line which can be found by dropping a vertical from the end of the line to the plane and joining it to the other end of the line. These lines form a right-angled triangle.
The angle is calculated using right-angled trigonometry. Pythagorasโ theorem may also be required.
Follow the worked example below
GCSE exam-style questions
- Calculate the size of angle ๐ต๐ท๐ธ.
Give the answer to one decimal place.
Angle ๐ต๐ท๐ธ = 19ยท1ยฐ, to one decimal place.
First identify the angle to be calculated.
Angle ๐ต๐ท๐ธ, marked as ฮธ, is the angle between ๐ธ๐ท and the plane ๐ด๐ต๐ถ๐ท.
To calculate angle ๐ต๐ท๐ธ two sides of the triangle must be known.
The lengths of ๐ด๐น and ๐ต๐ธ are equal, so ๐ต๐ธ = 5 cm.
Step 1. Calculate the length of ๐ต๐ท using Pythagorasโ theorem.
For triangle ๐ต๐ถ๐ท, Pythagorasโ theorem states ๐ถ๐ทยฒ + ๐ต๐ถยฒ = ๐ต๐ทยฒ. Substitute the values, ๐ถ๐ทยฒ = 8ยฒ and ๐ต๐ถยฒ = 12ยฒ, into the formula.
Calculate the value of the squares. 8ยฒ = 64, and 12ยฒ = 144. Add the squares together to get 208. This is the value of ๐ต๐ทยฒ.
Find ๐ต๐ท by calculating the square root of 208. Expressed as a surd ๐ต๐ท = โ208.
Step 2. Now two sides of the triangle ๐ต๐ท๐ธ are known, calculate angle ๐ต๐ท๐ธ using right-angled trigonometry.
In this triangle, the opposite (opp) and the adjacent (adj) are known. The trigonometric ratio needed must contain the opposite and the adjacent. The correct formula to use is the tangent ratio, tanฮธ = opp รท adj.
Substitute the values of opp, adj and ฮธ into the formula to form an equation. Here the opposite is 5, the adjacent is โ208 and the angle should be substituted with ฮธ.This gives tanฮธ = 5 รท โ208.
Work out the angle, ฮธ, using the inverse function of tangent.
Press โShift' then 'tan' to write tanโปยน on a scientific calculator.
Then type 5 รท โ208.
Remember to close the brackets.
This gives ฮธ = 19ยท1207โฆ Rounded to 1 decimal place, angle ๐ต๐ท๐ธ = 19ยท1ยฐ.
- Work out the angle between the line ๐ท๐น and the plane ๐ธ๐น๐บ๐ป.
Give the answer to one decimal place.
ฮธ = 28ยท3ยฐ, to one decimal place.
First identify the angle to be calculated.
Drop a vertical line from ๐ท to the plane. It meets the plane at point ๐ป. The line from ๐ป to ๐น is the projection of line ๐ท๐น. These lines form a right-angled triangle.
The angle to be calculated is ฮธ. In three letter notation this is angle ๐ท๐น๐ป. To calculate angle ฮธ, two sides of the triangle must be known.
Step 1. Calculate the length of ๐น๐ป using Pythagorasโ theorem.
For triangle ๐น๐บ๐ป, Pythagorasโ theorem states ๐น๐บยฒ + ๐บ๐ปยฒ = ๐น๐ปยฒ. The lengths of ๐น๐บ and ๐ด๐ท are equal, so ๐น๐บ = 5 cm. Substitute the values, ๐น๐บยฒ = 5ยฒ and ๐บ๐ปยฒ = 12ยฒ, into the formula.
Calculate the value of the squares. 5ยฒ = 25, and 12ยฒ = 144. Add the squares together to get 169. This is the value of ๐น๐ปยฒ.
Find ๐น๐ป, by calculating the square root of 169. ๐น๐ป = โ169 = 13 cm.
Step 2. Now two sides of the triangle ๐ท๐น๐ป are known, calculate angle ฮธ, using right-angled trigonometry.
In this triangle, the opposite (opp) and the adjacent (adj) are known. The trigonometric ratio needed must contain the opposite and the adjacent.
The correct formula to use is the tangent ratio, tanฮธ = opp รท adj.
Substitute the values of opp, adj and ฮธ into the formula to form an equation. Here the opposite is 7, the adjacent is 13 and the angle should be substituted with ฮธ. This gives tanฮธ = 7 รท 13
Work out the angle, ฮธ, using the inverse function of tangent.
Press 'Shift' then 'tan' to write tanโปยน on a scientific calculator.
Then type 7 รท 13.
Remember to close the brackets.
This gives ฮธ = 28ยท3007โฆ Rounded to 1 decimal place, ฮธ = 28ยท3ยฐ.
Check your understanding
Quiz โ 2D and 3D trigonometry problems
Practise what you know about 2D and 3D trigonometry problems with this quiz.
Now you've revised 2D and 3D trigonometry problems, why not look at right-angled trigonometry?
More on Geometry and measure
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