Key takeaway: This precalculus law sines cosines test is a printable PDF with eight questions on solving triangles. It checks area with sine, the Law of Sines, the ambiguous case, the Law of Cosines, Heron’s formula and bearings. Standard G-SRT.D.9 supplies the area rule. A calculator is required, and degree mode is essential. Show a sketch and the chosen law for each triangle. Expect about 45 minutes.
This test has 8 questions worth 20 points in total. Show every step, round only at the end, and state units where they apply. A scientific calculator in degree mode is allowed. The test takes about 45 minutes.
- Grade: Precalculus
- Chapter: Laws of Sines and Cosines and Triangle Area
- Common Core: G-SRT.D.9, G-SRT.D.10, G-SRT.D.11
- Time: 45 minutes
- Total: 20 points (Question 1: 2 pts; Question 2: 3 pts; Question 3: 3 pts; Question 4: 2 pts; Question 5: 2 pts; Question 6: 2 pts; Question 7: 3 pts; Question 8: 3 pts)
Precalculus Law Sines Cosines Test: questions 1–5
1. Area with sine (2 points)
Two sides of a triangle measure 14 cm and 9 cm and the angle between them is \(52^{\circ}\). Find the area to the nearest tenth of a square centimeter.
2. Law of Sines (3 points)
In triangle \(ABC\), \(A=38^{\circ}\), \(B=84^{\circ}\) and \(a=15\).
- Find \(C\). (1 point)
- Find \(b\) to the nearest hundredth. (1 point)
- Find \(c\) to the nearest hundredth. (1 point)
3. Ambiguous case (3 points)
In triangle \(ABC\), \(A=36^{\circ}\), \(a=14\) and \(b=20\).
- Show that two different triangles are possible. (1 point)
- Find both possible measures of \(B\) to the nearest tenth. (1 point)
- Find the two possible values of \(c\) to the nearest hundredth. (1 point)
4. Law of Cosines: find a side (2 points)
In triangle \(ABC\), \(a=13\), \(b=19\) and \(C=47^{\circ}\). Find \(c\) to the nearest hundredth.
5. Law of Cosines: find an angle (2 points)
A triangle has sides 10, 13 and 15. Find its largest angle to the nearest tenth of a degree.
Precalculus Law Sines Cosines Test: questions 6–8
6. Heron's formula (2 points)
Find the area of a triangle with sides 17, 25 and 28.
7. Ships and bearings (3 points)
Ship X leaves a port and sails 25 mi on bearing \(040^{\circ}\). Ship Y leaves the same port and sails 30 mi on bearing \(145^{\circ}\).
- Find the angle between the two paths. (1 point)
- Find the distance between the ships to the nearest hundredth of a mile. (2 points)
8. Lot area and height (3 points)
A triangular lot has sides 80 ft, 110 ft and 125 ft.
- Find the area to the nearest square foot. (2 points)
- Find the height to the 125 ft side to the nearest tenth of a foot. (1 point)
What do the early questions ask?
A triangle with sides 14 and 9 centimeters and an included \(52^{\circ}\) angle opens the paper. Then triangle \(ABC\) has \(A=38^{\circ}\), \(B=84^{\circ}\) and \(a=15\), so find \(C\) first and then the other sides. After that, the ambiguous case with \(A=36^{\circ}\), \(a=14\) and \(b=20\) asks you to show two triangles and solve one of them.
Which questions use the Law of Cosines?
Item 4 gives \(a=13\), \(b=19\) and \(C=47^{\circ}\), so the missing side is found by the cosine rule. Item 5 gives sides 10, 13 and 15 and wants the largest angle. Recall that the widest angle lies opposite the longest side. Item 6 applies Heron’s formula to 17, 25 and 28.
How do the last questions finish?
Two ships leave one port on different bearings, and you find their distance apart. The lot with sides 80, 110 and 125 feet needs an area and then a height. Because the area is known, the height follows from one half base times height. Include units on both answers.
Keep one minute per question for a sketch.
How can you prepare in advance?
Write a two-line decision rule: which data point to sines and which to cosines. Then test it on ten triangles from the worksheet. Quick, correct choices matter more than fast calculator work.
See every chapter on the Precalculus math page.
