Key takeaway: This precalculus law sines cosines worksheet offers 25 printable problems for solving non-right triangles. Problems 1 to 3 compute area with sine, problems 4 to 10 use the Law of Sines and the ambiguous case, problems 11 to 15 use the Law of Cosines, and problems 16 to 25 add Heron’s formula, bearings and surveying. Standard G-SRT.D.11 anchors the applications. The PDF includes a key, and a calculator in degree mode is needed.
These 25 problems practice triangle area with sine, the Law of Sines including the ambiguous SSA case, the Law of Cosines and Heron’s formula. They follow standards G-SRT.D.9, G-SRT.D.10 and G-SRT.D.11.
Problems run from easy to hard and finish with six applications involving bearings, surveying, flight paths and shorelines. Round sides to the nearest hundredth and angles to the nearest tenth unless told otherwise.
Precalculus Law Sines Cosines Worksheet: problems 1–5
1. Area from two sides and an angle
Find the area of each triangle with sides \(a\) and \(b\) and included angle \(C\). Round to the nearest hundredth when needed.
- \(a=10,\ b=7,\ C=30^{\circ}\)
- \(a=12,\ b=9,\ C=90^{\circ}\)
- \(a=8,\ b=5,\ C=150^{\circ}\)
- \(a=6,\ b=6,\ C=60^{\circ}\)
2. Area of a triangular sign
A triangular sign has two sides of 9 in. and 14 in. that meet at a \(38^{\circ}\) angle. Find its area to the nearest hundredth.
3. Find the angle from the area
A triangle has area 30 square units and two sides of lengths 10 and 12. Find every possible measure of the angle between those sides.
4. Law of Sines with two angles
In the triangle shown, \(A=40^{\circ}\), \(B=60^{\circ}\) and \(a=10\). Find \(C\), \(b\) and \(c\) to the nearest hundredth.
5. Find the third angle and both sides
In triangle \(ABC\), \(A=48^{\circ}\), \(C=71^{\circ}\) and \(c=20\). Solve the triangle. Round sides to the nearest hundredth.
Precalculus Law Sines Cosines Worksheet: problems 6–10
6. Law of Sines with the side between two angles
Sofia measures a triangular garden bed. Side \(a\) is 25 ft long and lies between angles \(B=52^{\circ}\) and \(C=67^{\circ}\). Find the other two sides to the nearest hundredth of a foot.
7. SSA with one solution
In triangle \(ABC\), \(A=70^{\circ}\), \(a=15\) and \(b=10\). Show that exactly one triangle exists, then find \(B\), \(C\) and \(c\) (angles to the nearest tenth, side to the nearest hundredth).
8. How many triangles?
Decide how many triangles satisfy each set of measurements.
- \(A=40^{\circ},\ a=5,\ b=10\)
- \(A=30^{\circ},\ a=5,\ b=10\)
- \(A=30^{\circ},\ a=7,\ b=10\)
- \(A=50^{\circ},\ a=12,\ b=10\)
9. Solve both triangles
In triangle \(ABC\), \(A=30^{\circ}\), \(a=7\) and \(b=10\). Find both possible triangles (angles to the nearest tenth, sides to the nearest hundredth).
10. Another ambiguous case
Priya knows \(A=42^{\circ}\), \(a=18\) and \(b=24\). Show that two triangles are possible and find the third side of each.
Precalculus Law Sines Cosines Worksheet: problems 11–15
11. Law of Cosines: SAS
In triangle \(ABC\), \(a=12\), \(b=7\) and \(C=55^{\circ}\). Find \(c\) to the nearest hundredth.
12. Law of Cosines with an obtuse angle
Two sides of a triangle measure 6 and 11 and the angle between them is \(120^{\circ}\). Find the third side to the nearest hundredth.
13. Law of Cosines: SSS
A triangle has sides 7, 9 and 12. Find its largest angle first, then the other two angles, to the nearest tenth of a degree.
14. Law of Cosines and the Pythagorean theorem
Apply the Law of Cosines to find the largest angle of each triangle.
- Sides 5, 12 and 13.
- Sides 8, 15 and 17.
- Explain what the result tells you about the Pythagorean theorem.
15. Choose the law
For each triangle, decide whether to start with the Law of Sines or the Law of Cosines. Give a reason.
- \(A=50^{\circ},\ B=70^{\circ},\ a=9\)
- \(b=8,\ c=11,\ A=40^{\circ}\)
- \(a=6,\ b=7,\ c=10\)
- \(A=35^{\circ},\ a=9,\ b=12\)
Precalculus Law Sines Cosines Worksheet: problems 16–20
16. Heron's formula
Use Heron’s formula to find the area of a triangle with sides 7, 8 and 9. Give the exact value and a decimal rounded to the nearest hundredth.
17. Heron's formula on a flat triangle
Find the area of the triangle with sides 9, 10 and 17.
18. Two ways to find an area
A triangle has sides 6, 8 and 10. Find its area with Heron’s formula, then with \(\frac{1}{2}ab\sin C\) using the angle between the sides 6 and 8, and compare.
19. Two boats leave a dock
At 1:00 p.m. two boats leave the same dock. Boat 1 travels at 24 mph on bearing \(035^{\circ}\) and boat 2 travels at 18 mph on bearing \(110^{\circ}\). How far apart are they at 3:00 p.m.? Round to the nearest hundredth.
20. Width of a lake
A surveyor stands at point \(C\) and measures 410 ft to one end of a lake and 560 ft to the other end. The angle between the two lines of sight is \(72^{\circ}\). Find the width of the lake to the nearest foot.
Precalculus Law Sines Cosines Worksheet: problems 21–25
21. A lot with two possible shapes
Noah measures an angle of \(28^{\circ}\) at corner \(A\) of a lot. The fence along \(AC\) is 90 ft, and the fence \(BC\) opposite \(A\) is 50 ft. Find both possible lengths of side \(AB\) to the nearest tenth of a foot.
22. Cost of sod
A triangular lawn has two sides of 120 ft and 150 ft meeting at a \(65^{\circ}\) corner. Sod costs $0.65 per square foot. Find the area to the nearest square foot and the cost of the sod to the nearest dollar.
23. Fertilizing a field
A triangular field has sides 300 ft, 400 ft and 500 ft. One bag of fertilizer covers 5,000 square feet and costs $18. Find the area with Heron’s formula, the number of bags needed and the total cost.
24. Plane flight on two bearings
A plane leaves an airport and flies 200 mi on bearing \(060^{\circ}\), then turns and flies 150 mi on bearing \(140^{\circ}\). Find the straight-line distance back to the airport (nearest tenth of a mile) and the bearing of the plane from the airport (nearest tenth of a degree).
25. Distance to a sailboat
Mei and Jamal stand 300 ft apart on a straight shore. Mei sights a sailboat at \(52^{\circ}\) from the shoreline and Jamal sights it at \(71^{\circ}\) from the shoreline, both measured toward the boat. Find the boat’s distance from Mei and its shortest distance from the shore, to the nearest tenth of a foot.
Which problems start the triangle solving?
To begin, a triangular sign in problem 2 has sides of 9 and 14 inches that meet at \(38^{\circ}\), and you find its area. Then problem 3 reverses the idea: an area of 30 with sides 10 and 12 fits two different angles, so list both. After that, angles of \(40^{\circ}\) and \(60^{\circ}\) with side 10 in problem 4 leave a third angle of \(80^{\circ}\). The Law of Sines gives the other sides.
How do the ambiguous-case problems differ?
In problem 7, exactly one triangle appears, because side \(a\) is longer than side \(b\). By contrast, problems 9 and 10 each produce two triangles. For \(A=30^{\circ}\), \(a=7\) and \(b=10\), the height 5 is below 7, and 7 is below 10. Meanwhile, problem 8 asks you to decide how many triangles fit, so compare \(a\) with \(b\sin A\) every time.
What do the cosine and Heron problems need?
Two sides and an included \(55^{\circ}\) angle start problem 11. Next, an obtuse \(120^{\circ}\) angle in problem 12 makes the cosine term negative. Because problem 13 gives sides 7, 9 and 12, find the largest angle first. Heron’s formula then handles 7, 8, 9 in problem 16 and the long flat triangle 9, 10, 17 in problem 17.
Where do the real-world problems occur?
Two boats leave a dock on different bearings in problem 19. Likewise, a surveyor measures a lake in problem 20. A 300-400-500 foot field fills problem 23, and a plane follows two bearings in problem 24.
How should you review this sheet?
List the given parts of each triangle beside the law you chose. Compare the pattern with the lesson. After that, redo any problem where the first choice was wrong, and test your answers with the quiz.
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Precalculus Laws of Sines and Cosines and Triangle Area worksheet
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