Law of Sines and Law of Cosines: math practice, Grade 12 – download the PDF

1 Solving an AAS triangle ★★★

In triangle \(ABC\), \(A=52^\circ\), \(B=71^\circ\) and \(a=18\) cm.

Find angle \(C\), then sides \(b\) and \(c\), rounded to the nearest tenth.

2 A missing angle ★★★

In triangle \(ABC\), \(A=48^\circ\), \(a=9\) in and \(b=7\) in. Since \(b

3 A side from SAS ★★★

In triangle \(ABC\), \(b=12\), \(c=15\) and \(A=60^\circ\). Find \(a\).

4 Area from two sides and the angle ★★★

A triangular sail has two edges of 10 m and 13 m that meet at an angle of \(30^\circ\). Find its area.

5 Heron’s formula ★★★

Find the exact area and a decimal approximation of a triangle with sides 7, 8 and 9.

6 True or false? ★★★

Decide whether each statement is true or false and justify.

1. If \(C=90^\circ\), the Law of Cosines reduces to \(c^2=a^2+b^2\).
2. If \(c^2>a^2+b^2\), then angle \(C\) is acute.
3. The Law of Sines alone can find the angles of a triangle whose three sides are known.

7 Which law first? ★★★

For each situation, state whether you start with the Law of Sines or the Law of Cosines, and why.

1. \(A=32^\circ\), \(B=81^\circ\), \(c=9\)
2. \(a=6\), \(b=9\), \(C=40^\circ\)
3. \(a=5\), \(b=7\), \(c=10\)
4. \(A=25^\circ\), \(a=8\), \(b=11\)

8 No triangle ★★★

Can a triangle have \(A=40^\circ\), \(a=6\) and \(b=10\)? Explain.

9 Two triangles ★★★

Let \(A=35^\circ\), \(a=8\) and \(b=11\). Show that two triangles exist and solve both, to the nearest tenth.

10 A right-triangle borderline ★★★

Let \(A=30^\circ\), \(a=5\) and \(b=10\). Solve the triangle and name its special feature.

11 Across the river ★★★

A surveyor wants the distance across a river to a tree T. She marks points A and B on her bank, 80 ft (about 24.4 m) apart, and measures \(\angle TAB=62^\circ\) and \(\angle TBA=71^\circ\). How far is the tree from A, and from B?

12 All angles from three sides ★★★

A triangle has sides \(a=5\), \(b=7\) and \(c=9\). Find all three angles to the nearest hundredth of a degree.

13 Area with two angles and a side ★★★

In triangle \(ABC\), \(A=50^\circ\), \(B=60^\circ\) and \(c=10\). Find the area.

14 Diagonals of a parallelogram ★★★

A parallelogram has sides of 8 cm and 11 cm and one angle of \(70^\circ\). Find the lengths of both diagonals.

15 A ship’s return trip ★★★

A ship leaves port P and sails 30 mi on bearing \(050^\circ\) to Q, then 45 mi on bearing \(125^\circ\) to R (1 mi is about 1.61 km). Find the distance PR and the bearing the ship should follow to return directly from R to P.

16 Spotting a fire ★★★

Two lookout towers A and B stand 5 km apart on an east-west line, B being east of A. The ranger at A sees a fire on bearing \(048^\circ\); the ranger at B sees it on bearing \(330^\circ\).

Find the distances AF and BF, and how far north of line AB the fire is.

17 Heron, altitude and inscribed circle ★★★

A triangle has sides 17, 25 and 28.

1. Find its area.
2. Find its shortest altitude.
3. The radius of the inscribed circle is \(r=\dfrac{K}{s}\). Find \(r\).

18 A land parcel ★★★

A quadrilateral parcel ABCD has \(AB=50\) m, \(BC=70\) m, \(CD=45\) m, \(DA=55\) m and \(\angle B=60^\circ\). Find its total area. (Cut it along diagonal AC.)

19 Recognizing a 120 degree angle ★★★

A triangle has sides 7, 8 and 13. Show that its largest angle is exactly \(120^\circ\), then find the exact area in two ways.

20 How many triangles for each side? ★★★

Angle \(A=30^\circ\) and side \(b=12\) are fixed. For which values of \(a\) does the triangle exist in 0, 1 or 2 versions?

21 The circumscribed circle ★★★

For the triangle with sides 13, 14 and 15 (area 84), use \(R=\dfrac{abc}{4K}\) to find the circumradius. Then use \(\dfrac{a}{\sin A}=2R\) to find \(\sin A\) for the angle opposite 13.