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.Β :
Area of a triangle = \(\frac{1}{2}\)ππ sinπΆ
In the formula π and π are the lengths of the two known sides and πΆ is the size of the angle between them.
Follow the worked example below
GCSE exam-style questions
- Calculate the area of triangle πππ .
Give the answer to 1 decimal place.
Area = 36Β·8 cmΒ² to 1 decimal place.
- Label the sides of the triangle. Since the vertices are not called π΄, π΅ and πΆ let vertex, π, with angle 130Β° be πΆ.
The choice of π΄ and π΅ does not matter. Let vertex π be π΄ and vertex π be π΅.
The 12 cm side, opposite angle π΄, is called π. The 8 cm side, opposite angle π΅, is called π.
- Substitute the values of π, π and πΆ into the formula to give
Area = \(\frac{1}{2}\) Γ 12 Γ 8 Γ sin(130)
- Type \(\frac{1}{2}\) Γ 12 Γ 8 Γ 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 = 36Β·7701β¦
Therefore, rounded to one decimal place, Area = 36Β·8 cmΒ².
- Calculate the area of the parallelogram.
Give the answer to 1 decimal place.
Area = 63Β·6 mΒ² 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 130Β° be πΆ.
The choice of π΄ and π΅ does not matter.
The 5 m side, opposite angle π΄, is called π.
The 13 m side, opposite angle π΅, is called π.
- Substitute the values of π, π and πΆ into the formula to give
Area = \(\frac{1}{2}\) Γ 12 Γ 8 Γ sin(130)
- Type \(\frac{1}{2}\) Γ 5 Γ 13 Γ 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 = 31Β·7897β¦
- Find the area of the parallelogram by multiplying this by two.
31Β·7897β¦ Γ 2 = 63Β·5795β¦
Therefore, rounded to one decimal place, the area of the parallelogram = 63Β·6 mΒ².
- The area of the triangle is 30 cmΒ².
Calculate the size of the acute angle π₯ to 1 decimal place.
π₯ = 59Β·0Β° to 1 decimal place.
Label the sides of the triangle. The 10 cm side, opposite angle π΄, is called π. The 7 cm side, opposite angle π΅, is called π.
Substitute the values of π, π, πΆ and the area into the formula to give 30 = \(\frac{1}{2}\) Γ 10 Γ 7 Γ sin(π₯)
\(\frac{1}{2}\) Γ 7 Γ 10 = 35. This simplifies to 30 = 35 Γ sin(π₯).
Find sin(π₯) by dividing both sides by 35. This gives 30 Γ· 35 = sin(π₯).
Work out angle, π₯, by using the inverse function of sine.
π₯ = sinβ»ΒΉ (30/35).
- To write sinβ»ΒΉ on a scientific calculator, press 'Shift' then 'sin'. Then type 30 Γ· 35. Remember to close the brackets.
This gives π₯ = 58Β·9972β¦
Rounded to 1 decimal place, π₯ = 59Β·0Β°.
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 π, π΄ and π΅
- an angle when two sides and an opposite angle are known, eg π, π΄ and π
Use the cosine rule on any triangle to calculate:
- a side when two sides and the included angle are known, eg π, π and πΆ
- an angle when all three sides are known
- 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 π· to πΈ and then from πΈ to πΉ.
πΈ is 25 miles from π·, on a bearing of 050Β°.
πΉ is 32 miles from πΈ, on a bearing of 105Β°.
Work out the direct distance from π· to πΉ.
π·πΉ = 50Β·7 miles, to one decimal place.
- First calculate the other angles using geometry. The two north lines are parallel.
Co-interior angles add up to 180Β°. The angle to the left of πΈ = 180 β 50 = 130Β°.
Use angles at a point to calculate angle π·πΈπΉ.
Angle π·πΈπΉ = 360 β 105 β 130 = 125Β°.
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 π΄, π΅ and πΆ, so rename the vertex with angle 125Β° as π΄.
The choice of π΅ and π does not matter. Let π· be π΅ and πΉ be πΆ.
The 25-mile side, opposite angle πΆ, is called π.
The 32-mile side, opposite angle π΅, is called π.
The side π΅π, opposite angle π΄, is called π.
- Substitute the values of π΄, π, π and π into the formula to give π·πΉΒ² = 32Β² + 25Β² β (2 Γ 32 Γ 25) cos(125).
32Β² = 1024 , 25Β² = 625 and 2 Γ 32 Γ 25 = 1600, so this simplifies to π·πΉΒ² = 1024 + 625 β 1600cos(125).
- Type 1024 + 625 β 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 π·πΉΒ² = 2566Β·7222β¦
It is important not to round the number at this stage.
- Find π·πΉ by calculating the square root of 2566Β·7222β¦
Type the square root button followed by the βAnsβ button into a scientific calculator.
This gives the answer of π·πΉ = 50Β·6628β¦. Therefore, rounded to one decimal place, π·πΉ = 50Β·7 miles.
- Quadrilateral ππππ is formed by combining two triangles.
Work out the length of ππ. Give the answer to one decimal place.
ππ = 13Β·4 cm, to one decimal place.
Split the quadrilateral into two separate triangles and apply non-right angled trigonometry.
Step 1. Calculate the length of ππ using the sine rule.
- Label the sides of the triangle. Since the vertices are not called π΄, π΅ and πΆ let vertex π be π΄, π be π΅ and π be πΆβ.
The 12 cm side, opposite angle π΅, is called π. The side ππ or π΄π΅, opposite angle πΆ, is called π, and the unknown side, opposite angle π΄, is called π.
Since neither side π, or angle π΄ is known this is the portion of the sine rule formula that will not be used.
Substitute the values of π΅, πΆ, π and π into the formula to give \(\frac{12}{sin(66)}\) = \(\frac{π΄π΅}{sin(97)}\).
Rearrange the equation to make π΄π΅ the subject. Find the value of π΄π΅ by multiplying both sides of the equation by sin(97). This gives \(\frac{12sin(97)}{sin(66)}\) = π΄π΅.
Type 12sin(97) Γ· sin(66) into a scientific calculator. This gives π΄π΅ = 13Β·0377 β¦ It is important not to round the number at this stage.
Step 2. Calculate the length of ππ using the cosine rule.
- Label the sides of the triangle. Keep π as π΄, π as π΅ and let π be πΆβ.
The 9 cm side, opposite angle π΅, is called π. The side, opposite angle πΆβ, is called π, and is the answer that was calculated in Step 1.
The side labelled ππ, opposite angle π΄, is called π.
- Substitute the values of π΄, π, π and π into the formula to give ππΒ² = 92 + ππΒ² - (2 Γ 10 Γ ππ) cos(72).
92 = 81 and 2 Γ 9 Γ ππ = 18ππ, so this simplifies to πΒ² = 81+ ππΒ² β 18ππcos(72).
- Type 81 + ππΒ² β 18ππcos(72) into a scientific calculator. Use the βAnsβ button each time you need ππ.
This gives ππΒ² = 178Β·4642β¦
It is again important not to round the number at this stage.
- Find ππ, by calculating the square root of 178Β·4642β¦
Type the square root button followed by the βAnsβ button into a scientific calculator.
This gives the answer of ππ = 13Β·3589β¦.
Therefore, rounded to one decimal place, ππ = 13Β·4 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. Pythagorasβ theorem may also be required.
Follow the worked example below
GCSE exam-style questions
- Calculate the size of angle π΅π·πΈ.
Give the answer to one decimal place.
Angle π΅π·πΈ = 19Β·1Β°, to one decimal place.
First identify the angle to be calculated.
Angle π΅π·πΈ, marked as ΞΈ, is the angle between πΈπ· and the plane π΄π΅πΆπ·.
To calculate angle π΅π·πΈ two sides of the triangle must be known.
The lengths of π΄πΉ and π΅πΈ are equal, so π΅πΈ = 5 cm.
Step 1. Calculate the length of π΅π· using Pythagorasβ theorem.
For triangle π΅πΆπ·, Pythagorasβ theorem states πΆπ·Β² + π΅πΆΒ² = π΅π·Β². Substitute the values, πΆπ·Β² = 8Β² and π΅πΆΒ² = 12Β², into the formula.
Calculate the value of the squares. 8Β² = 64, and 12Β² = 144. Add the squares together to get 208. This is the value of π΅π·Β².
Find π΅π· by calculating the square root of 208. Expressed as a surd π΅π· = β208.
Step 2. Now two sides of the triangle π΅π·πΈ are known, calculate angle π΅π·πΈ 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, tanΞΈ = opp Γ· adj.
Substitute the values of opp, adj and ΞΈ into the formula to form an equation. Here the opposite is 5, the adjacent is β208 and the angle should be substituted with ΞΈ.This gives tanΞΈ = 5 Γ· β208.
Work out the angle, ΞΈ, using the inverse function of tangent.
Press βShift' then 'tan' to write tanβ»ΒΉ on a scientific calculator.
Then type 5 Γ· β208.
Remember to close the brackets.
This gives ΞΈ = 19Β·1207β¦ Rounded to 1 decimal place, angle π΅π·πΈ = 19Β·1Β°.
- Work out the angle between the line π·πΉ and the plane πΈπΉπΊπ».
Give the answer to one decimal place.
ΞΈ = 28Β·3Β°, to one decimal place.
First identify the angle to be calculated.
Drop a vertical line from π· to the plane. It meets the plane at point π». The line from π» to πΉ is the projection of line π·πΉ. These lines form a right-angled triangle.
The angle to be calculated is ΞΈ. In three letter notation this is angle π·πΉπ». To calculate angle ΞΈ, two sides of the triangle must be known.
Step 1. Calculate the length of πΉπ» using Pythagorasβ theorem.
For triangle πΉπΊπ», Pythagorasβ theorem states πΉπΊΒ² + πΊπ»Β² = πΉπ»Β². The lengths of πΉπΊ and π΄π· are equal, so πΉπΊ = 5 cm. Substitute the values, πΉπΊΒ² = 5Β² and πΊπ»Β² = 12Β², into the formula.
Calculate the value of the squares. 5Β² = 25, and 12Β² = 144. Add the squares together to get 169. This is the value of πΉπ»Β².
Find πΉπ», by calculating the square root of 169. πΉπ» = β169 = 13 cm.
Step 2. Now two sides of the triangle π·πΉπ» are known, calculate angle ΞΈ, 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, tanΞΈ = opp Γ· adj.
Substitute the values of opp, adj and ΞΈ into the formula to form an equation. Here the opposite is 7, the adjacent is 13 and the angle should be substituted with ΞΈ. This gives tanΞΈ = 7 Γ· 13
Work out the angle, ΞΈ, using the inverse function of tangent.
Press 'Shift' then 'tan' to write tanβ»ΒΉ on a scientific calculator.
Then type 7 Γ· 13.
Remember to close the brackets.
This gives ΞΈ = 28Β·3007β¦ Rounded to 1 decimal place, ΞΈ = 28Β·3Β°.
Check your understanding
Quiz β 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
Find out more by working through a topic
- count34 of 35
- count35 of 35
- count1 of 35
- count2 of 35
