Triangle problems can involve using a combination of different rules concerning area, trigonometry and Pythagoras' theorem. You will need to know how to combine different equations and rules in order to find missing sides and angles in triangles.
Equations
Rule
equations
Trigonometric ratios
sin(θ)=hypotenuseopposite
cos(θ)=hypotenuseadjacent
tan(θ)=adjacentopposite
Area of a triangle
Area=21bh
Area=21absin(C)
The cosine rule
a2=b2+c2−2bccos(A)
cos(A)=2bcb2+c2−a2
The sine rule
sin(A)a=sin(B)b=sin(C)c
asin(A)=bsin(B)=csin(C)
Pythagoras' theorem
a2+b2=c2
Triangle problems
There are a variety of triangle rules and equations you will need to use to find a missing angle or side. You will also need to be able to combine these equations algebraically to calculate unknown values.
Example 1
In the triangle below, find the values of x, y and z to one decimal place.
△ABC has three sides given which can be used to find a missing angle using the cosine rule:
In the triangle below, given that ADC is a straight line, find m to one decimal place.
△BDC has two sides which are the same length and an angle which is not 60∘. So, it must be an isosceles triangle:
∠DBC=∠BCD=55∘
Angles in a triangle add up to 180∘:
∠BDC=180−55−55=70°
ADC is a straight line. Angles on a straight line add up to 180∘:
∠ADB=180−70=110∘
In △ADB, two sides have been given: BA and BD. An angle has also been found. This means that the cosine rule can be used to find the missing side:
a2=b2+c2−2bccos(A)
Substitute values into the cosine rule where ∠ADB is the focus:
(8)2=(5)2+m2−2(5)mcos(110)
0=x2−2(5)mcos(110)+(5)2−(8)2
m2−10cos(110)(m)−39=0
Solve the quadratic to find m:
m=2(1)−(10cos(110))±(−10cos(110))2−4(1)(−39)
mm=−4.764...=8.185...
mis a side length so it must be a positive value:
m=8.2cm(1d.p.)
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Unit 1
Finding missing lengths in a triangle
Unit 2
Sine and cosine rules - Higher
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Unit 3
Solving triangle problems
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FAQs - Frequently Asked Questions
What is the sine rule used for?
The sine rule is used to find a missing angle given two sides and one opposite angle or to find a missing angle given two sides and one opposite angle.
What is the angle form of the cosine rule used for?
The angle form of the cosine rule is used to find an unknown angle given all three sides of a triangle.
What is Pythagoras' theorem used for?
Pythagoras' theorem is used to calculate the length of a missing side in a right-angled triangle when given the lengths of the other two sides.