Precalculus Law of Sines and Law of Cosines Worksheet (PDF)

Precalculus math worksheet: Law of Sines and Law of Cosines – Maths PDF

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This laws sines cosines worksheet for Precalculus covers law of sines and law of cosines (Common Core G-SRT.D.10, G-SRT.D.11, G-SRT.C.8), with a free PDF to print.

These 20 problems practice the Law of Sines, the Law of Cosines and triangle area (Common Core G-SRT.D.10, G-SRT.D.11 and G-SRT.C.8). You will solve right and oblique triangles, examine the ambiguous case and use surveying and navigation settings.

Precalculus · Common Core State Standards: G-SRT.D.10, G-SRT.D.11, G-SRT.C.8

Laws sines cosines worksheet: exercises 1 to 10

Exercise 1 – Right triangle: find a leg

In a right triangle, the hypotenuse is \(20\) and one acute angle is \(35^{\circ}\). Find the side opposite that angle.

Exercise 2 – Right triangle: tangent

A \(62^{\circ}\) angle in a right triangle has an adjacent side of \(8\) ft. Find the opposite side.

Exercise 3 – Solve a right triangle

A right triangle has legs \(7\) and \(24\). Find the hypotenuse and both acute angles to the nearest hundredth.

Exercise 4 – Law of Sines (AAS)

Triangle with two angles and a side.

Solve the triangle with \(A=40^{\circ}\), \(B=60^{\circ}\), \(a=10\). Round sides to the nearest hundredth.

Exercise 5 – Law of Sines (ASA-like)

Solve the triangle with \(A=50^{\circ}\), \(C=70^{\circ}\), \(c=12\).

Exercise 6 – SSA with one triangle

Solve the triangle with \(a=15\), \(b=10\), \(A=40^{\circ}\).

Exercise 7 – Ambiguous case: two triangles

Two triangles fit the data.

Triangle \(ABC\) has \(A=35^{\circ}\), \(a=7\) and \(b=10\). Find both possible values of \(B\) and the number of triangles.

Exercise 8 – No triangle

Does a triangle exist with \(A=50^{\circ}\), \(a=5\) and \(b=10\)?

Exercise 9 – Law of Cosines (SAS)

Find \(c\) if \(a=8\), \(b=11\) and \(C=48^{\circ}\).

Exercise 10 – Law of Cosines (SSS)

Find the largest angle of a triangle with sides \(7\), \(9\) and \(12\).

Exercises 11 and up

Exercise 11 – All three angles

Find all three angles of the triangle with sides \(5\), \(6\) and \(7\).

Exercise 12 – Area from two sides

Find the area of a triangle with sides \(9\) and \(14\) and included angle \(35^{\circ}\).

Exercise 13 – Heron's formula

Find the area of a triangle with sides \(13\), \(14\) and \(15\).

Exercise 14 – Choose the law

Tell which law to use first: (a) \(a=5\), \(b=8\), \(C=70^{\circ}\); (b) \(A=30^{\circ}\), \(B=80^{\circ}\), \(c=9\); (c) \(a=4\), \(b=6\), \(c=9\); (d) \(a=7\), \(b=5\), \(A=50^{\circ}\).

Exercise 15 – Width of a pond

Maya’s survey of a pond.

Maya stands at point \(C\), \(120\) ft from point \(A\) and \(95\) ft from point \(B\) on opposite sides of a pond. The angle \(ACB\) is \(62^{\circ}\). Find the distance \(AB\) to the nearest tenth.

Exercise 16 – Two ships

Positions of two ships.

Ship A is \(12\) mi from port on bearing \(040^{\circ}\). Ship B is \(9\) mi from port on bearing \(115^{\circ}\). How far apart are the ships to the nearest tenth of a mile?

Exercise 17 – Height of a tree

Angles of elevation to the top of a tree.

From a point on level ground the angle of elevation to the top of a tree is \(30^{\circ}\). After walking \(40\) ft toward the tree, the angle is \(45^{\circ}\). Find the height \(h\) of the tree to the nearest tenth of a foot.

Exercise 18 – Triangular garden

A triangular garden has sides \(18\) ft, \(24\) ft and \(30\) ft. Find its area with Heron’s formula and find its largest angle.

Exercise 19 – Roof truss

A roof truss is an isosceles triangle with two sides of \(10\) ft and a base of \(12\) ft. Find the apex angle and the base angles to the nearest hundredth.

Exercise 20 – Area from angles and a side

A triangle has \(a=10\), \(A=40^{\circ}\) and \(B=70^{\circ}\). Find its area to the nearest tenth.

Key facts behind this laws sines cosines worksheet

Key facts

Law of Sines: a/sin A = b/sin B = c/sin C.
Law of Cosines: c^2 = a^2 + b^2 – 2ab cos C.
SSA can give 0, 1 or 2 triangles (the ambiguous case).
Area = (1/2) a b sin C; Heron: sqrt(s(s – a)(s – b)(s – c)) with s = (a + b + c)/2.
The angles of a triangle add to 180 degrees; the largest angle is opposite the longest side.

How to use this laws sines cosines worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the laws sines cosines worksheet as a PDF with the button below and keep it with your notes.

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