Laws of Sines and Cosines and Triangle Area: Precalculus Math Quiz – PDF Download
Ten multiple-choice questions on the area formula with sine, the Law of Sines, the ambiguous case, the Law of Cosines, Heron’s formula and bearings. Choose the best answer, then read the explanation to see the common mistake.
🎯 Laws of Sines and Cosines and Triangle Area – Precalculus math quiz
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Question 1
What is the area of the triangle shown?Sides 10 and 12 with a 30 degree angle between them.
A\(60\)
B\(30\)
C\(52.0\)
D\(103.9\)
Question 2
A triangle has \(a=9\), \(b=14\) and \(C=38^{\circ}\). Which tool should you use first to find \(c\)?
ALaw of Sines
BHeron's formula
CPythagorean theorem
DLaw of Cosines
Question 3
In the triangle shown, what is side \(b\) to the nearest hundredth?Two angles and the side opposite A.
A\(18.96\)
B\(7.59\)
C\(11.04\)
D\(20.60\)
Question 4
How many triangles have \(A=30^{\circ}\), \(a=4\) and \(b=10\)?
A1
B0
C2
D3
Question 5
Which equation is the Law of Cosines for side \(c\)?
A\(c^{2}=a^{2}+b^{2}+2ab\cos C\)
B\(c^{2}=a^{2}+b^{2}-2ab\sin C\)
C\(c^{2}=a^{2}+b^{2}-ab\cos C\)
D\(c^{2}=a^{2}+b^{2}-2ab\cos C\)
Question 6
Use Heron's formula to find the area of a triangle with sides 5, 6 and 7.
A\(216\)
B\(7.35\)
C\(29.39\)
D\(14.70\)
Question 7
A ship sails on a bearing of \(135^{\circ}\). In which direction is it heading?
Asoutheast
Bnortheast
Csouthwest
Dnorthwest
Question 8
Two paths leave a point on bearings \(050^{\circ}\) and \(130^{\circ}\), as shown. What is the angle between the paths?Bearings are measured clockwise from north.
A\(180^{\circ}\)
B\(100^{\circ}\)
C\(80^{\circ}\)
D\(40^{\circ}\)
Question 9
A triangle has area 24 square feet and sides of 8 ft and 10 ft. What is the acute angle between those sides?
A\(53.1^{\circ}\)
B\(143.1^{\circ}\)
C\(36.9^{\circ}\)
D\(30^{\circ}\)
Question 10
How many triangles have \(A=40^{\circ}\), \(a=8\) and \(b=10\)?
A2
B0
C1
D3
🎉 Quiz complete!
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Remember:
Standards G-SRT.D.9, G-SRT.D.10 and G-SRT.D.11: triangle area with sine, the Law of Sines and the Law of Cosines in applied problems.
Area: \(\frac{1}{2}ab\sin C\); Heron: \(\sqrt{s(s-a)(s-b)(s-c)}\) with \(s=\frac{a+b+c}{2}\).
Law of Sines: \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\); use it for AAS, ASA and SSA.
Law of Cosines: \(c^{2}=a^{2}+b^{2}-2ab\cos C\); use it for SAS and SSS.
SSA can give 0, 1 or 2 triangles, depending on how \(a\) compares with \(b\sin A\) and \(b\).
