In triangle ABC, a = 3, b = 5 and γ = 120°. Find the side c.
- Law of cosines: c2 = a2 + b2 − 2ab cos γ = 9 + 25 − 2 × 3 × 5 × cos 120°.
- cos 120° = −1/2, so −2 × 3 × 5 × (−1/2) = +15.
- c2 = 34 + 15 = 49, so c = 7.
Answer: c = 7.
Solving triangles
Maths · Application of trigonometry. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
In triangle ABC the sides a, b, c are opposite the angles α, β, γ. A triangle has six elements. To solve it means to find the unknown elements from three given ones, at least one of which is a side.
A right-angled triangle needs only the basic ratios. An angle of elevation or depression is always measured from the horizontal.
An oblique triangle has no right angle. Use the law of cosines when you know two sides and the included angle, or all three sides. Use the law of sines when you know a side and its opposite angle. The law of tangents and the half-angle formulas are other ways to find angles from sides.
Rules to remember
- α + β + γ = 180°
- Law of cosines: a2 = b2 + c2 − 2bc cos α, so cos α = (b2 + c2 − a2)/(2bc); similarly for b and c
- Law of sines: a/sin α = b/sin β = c/sin γ
- Law of tangents: (a − b)/(a + b) = tan((α − β)/2) ÷ tan((α + β)/2)
- With s = (a + b + c)/2: sin(α/2) = √((s − b)(s − c)/(bc)) and cos(α/2) = √(s(s − a)/(bc))
- tan(α/2) = √((s − b)(s − c)/(s(s − a)))
- Right-angled triangle: height = horizontal distance × tan(angle of elevation)
Common mistake
In the law of cosines the angle must be the one between the two sides you multiply, and when that angle is obtuse its cosine is negative, so the last term is added.
Worked example
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