Worksheets · Precalculus

Precalculus law of sines and cosines worksheet

The two rules for triangles without a right angle. Sines pairs a side with its opposite angle; cosines is the Pythagorean theorem with a correction term for the angle not being 90 degrees.

Select your difficulty

Pick more than one for a sheet that mixes them.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Laws of Sines and Cosines

Date Period

Find the missing measure.

  1. In a triangle, A=45∘, C=105∘ and a=27 ext.FindthesideoppositeB
  2. In a triangle, a=70, b=69 and C=120∘. Find c
  3. In a triangle, b=40, c=51 and a=79. Find A
  4. In a triangle, A=120∘, C=30∘ and a=40 ext.FindthesideoppositeB
  5. In a triangle, a=35, b=46 and C=120∘. Find c
  6. In a triangle, a=35, b=48 and c=43. Find C
  7. In a triangle, A=30∘, C=90∘ and a=49 ext.FindthesideoppositeB
  8. In a triangle, b=37, c=31 and A=60∘. Find a
  9. In a triangle, a=11, b=85 and c=91. Find C
  10. In a triangle, A=45∘, C=75∘ and a=46 ext.FindthesideoppositeB
  11. In a triangle, c=61, a=37 and B=120∘. Find b
  12. In a triangle, c=11, a=85 and b=91. Find B
  13. In a triangle, A=45∘, C=75∘ and a=48 ext.FindthesideoppositeB
  14. In a triangle, a=43, b=46 and C=60∘. Find c
  15. In a triangle, b=16, c=55 and a=49. Find A
  16. In a triangle, A=30∘, C=105∘ and a=34 ext.FindthesideoppositeB
  17. In a triangle, a=55, b=65 and C=60∘. Find c
  18. In a triangle, b=16, c=63 and a=65. Find A
  19. In a triangle, A=120∘, C=30∘ and a=53 ext.FindthesideoppositeB
  20. In a triangle, c=67, a=64 and B=90∘. Find b

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Laws of Sines and CosinesAnswer keyVersion 1

Date Period

Find the missing measure.

  1. In a triangle, A=45∘, C=105∘ and a=27 ext.FindthesideoppositeB2722
  2. In a triangle, a=70, b=69 and C=120∘. Find c14491
  3. In a triangle, b=40, c=51 and a=79. Find A120∘
  4. In a triangle, A=120∘, C=30∘ and a=40 ext.FindthesideoppositeB4033
  5. In a triangle, a=35, b=46 and C=120∘. Find c4951
  6. In a triangle, a=35, b=48 and c=43. Find C60∘
  7. In a triangle, A=30∘, C=90∘ and a=49 ext.FindthesideoppositeB493
  8. In a triangle, b=37, c=31 and A=60∘. Find a137
  9. In a triangle, a=11, b=85 and c=91. Find C120∘
  10. In a triangle, A=45∘, C=75∘ and a=46 ext.FindthesideoppositeB236
  11. In a triangle, c=61, a=37 and B=120∘. Find b7347
  12. In a triangle, c=11, a=85 and b=91. Find B120∘
  13. In a triangle, A=45∘, C=75∘ and a=48 ext.FindthesideoppositeB246
  14. In a triangle, a=43, b=46 and C=60∘. Find c1987
  15. In a triangle, b=16, c=55 and a=49. Find A60∘
  16. In a triangle, A=30∘, C=105∘ and a=34 ext.FindthesideoppositeB342
  17. In a triangle, a=55, b=65 and C=60∘. Find c353
  18. In a triangle, b=16, c=63 and a=65. Find A90∘
  19. In a triangle, A=120∘, C=30∘ and a=53 ext.FindthesideoppositeB5333
  20. In a triangle, c=67, a=64 and B=90∘. Find b8585

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

In a triangle, a=7, b=5 and C=60∘. Find c

  1. The law of cosines: c squared equals a squared plus b squared minus 2ab cos C.
  2. 49 plus 25 minus 2(7)(5)(one half).
  3. That is 39.

The answer is 39.

Where students go wrong

Choosing the wrong rule. If you have a side and the angle opposite it, use sines; if you have two sides and the angle between them, only cosines will work. The other slip is forgetting that the correction term is subtracted, not added.

Also called sine rule, cosine rule, non-right triangles or oblique triangles.

What you can put on this worksheet

The law of sines
In a triangle, A=45∘, B=30∘ and a=6. Find b → 32
A missing side, from two sides and the angle between
In a triangle, a=5, b=7 and C=120∘. Find c → 109
A missing angle, from three sides
In a triangle, c=7, a=15 and b=13. Find B → 60∘

Questions about these worksheets

Yes. Every law of sines and cosines sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the law of sines and cosines questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 3: the law of sines, a missing side, from two sides and the angle between and a missing angle, from three sides. Tick as many as you want and set how many of each, or let it spread them evenly.

Choosing the wrong rule. If you have a side and the angle opposite it, use sines; if you have two sides and the angle between them, only cosines will work. The other slip is forgetting that the correction term is subtracted, not added.