The Law of Sines & Cosines — Practice
Two rules extend trigonometry to every triangle, not just the right-angled ones.
What these questions cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Ratio — A comparison of two quantities, like 3 cups flour to 2 cups water.
- Area — Space inside, measured in square units.
- Included angle — The angle between two named sides. The A in SAS must be included.
- Sine — sin θ = oppositehypotenuse.
- Law of sines — asin A = bsin B = csin C.
- Law of cosines — c2 = a2 + b2 - 2abcos C.
Problem 1
In triangle ABC, A = 40° and B = 75°. Find C.
Worked answer
65
180-40-75 = 65
Problem 2
Two angles and one side are known. Which rule applies?
- Law of sines
- Law of cosines
- Pythagoras
- Neither
Worked answer
Law of sines
The third angle follows, giving a complete pair
Problem 3
All three sides are known and no angles. Which rule applies?
- Law of cosines
- Law of sines
- Pythagoras
- Neither
Worked answer
Law of cosines
No angle is known, so the sine rule cannot start
Problem 4
In triangle ABC, A = 30°, a = 10 and B = 45°. Find b to one decimal place.
Worked answer
14.1
b = 10 sin 45°sin 30°
= 7.0710.5 ≈ 14.1
Problem 5
Can a triangle have sides 3, 4 and 8?
- No
- Yes
- Only if it is obtuse
- Only if it is right-angled
Worked answer
No
3+4 = 7, which is less than 8
The short sides cannot reach across
Problem 6
In triangle ABC, A = 50°, B = 60° and a = 8. Find b to one decimal place.
Worked answer
9.0
b = 8sin 60°sin 50°
= 6.9280.766 ≈ 9.0
Problem 7
A triangle has sides 5 and 6 with an included angle of 60°. Find the third side to one decimal place.
Worked answer
5.6
25+36-2(5)(6)(0.5)
= 61-30 = 31
c = √31 ≈ 5.6
Problem 8
A triangle has sides 2 and 3 with an included angle of 120°. Find the third side to one decimal place.
Worked answer
4.4
4+9-2(2)(3)(-0.5)
= 13+6 = 19
c = √19 ≈ 4.4
Problem 9
A triangle has sides 4, 5 and 6. Find the angle opposite the side of length 6, to the nearest degree.
Worked answer
83
36 = 16+25-2(4)(5)cos C
cos C = 540 = 0.125
C ≈ 83°
Problem 10
A triangle has sides 3, 4 and 5. Find the largest angle in degrees.
Worked answer
90
9+16 = 25
So it is right-angled
Problem 11
Find the area of a triangle with sides 6 and 8 and an included angle of 30°.
Worked answer
12
12(6)(8)(0.5)
= 12
Problem 12
In triangle ABC, a = 8, A = 40° and b = 10. Find the acute value of B to the nearest degree.
Worked answer
53
sin B = 10sin 40°8 = 0.8035
B ≈ 53°
Problem 13
With a = 5, b = 8 and A = 30°, how many triangles are possible?
- Two
- One
- None
- Three
Worked answer
Two
sin B = 8(0.5)5 = 0.8, so B = 53° or 127°
30+53 and 30+127 are both under 180, so both work
Problem 14
A triangle has sides 7, 8 and 12. Find the largest angle to the nearest degree.
Worked answer
106
144 = 49+64-2(7)(8)cos C
112cos C = 113-144 = -31
cos C ≈ -0.277, so C ≈ 106°
Problem 15
Two observers 100 m apart both sight a tree, at 40° and 65° from the line joining them. Find the distance from the first observer to the tree, to one decimal place.
Worked answer
93.8
Third angle = 180-40-65 = 75°
d = 100sin 65°sin 75° = 90.6396.59 × 100 ≈ 93.8
Problem 16
A triangle has sides 7 and 9 with an included angle of 90°. Find the third side to one decimal place.
Worked answer
11.4
49+81 = 130
c = √130 ≈ 11.4
Exactly Pythagoras