Worksheets · Precalculus

Trig ratios of general angles worksheet for Precalculus

Precalculus defines the ratios from any point (x, y) on the terminal side, with r = √(x² + y²), so every quadrant and all six functions work the same way. Some questions ask for one ratio, others for all six, and answers with radicals are rationalized.

Also taught in Algebra 2, at its own level and with its own examples.

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Pick more than one for a sheet that mixes them.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Trig Ratios of General Angles

Date Period

Find each ratio.

  1. Find a ratio from a point

    The point (60,11) is on the terminal side of θ. Find tan⁡θ.
  2. The point (−5,12) is on the terminal side of θ. Find tan⁡θ.
  3. The point (48,−55) is on the terminal side of θ. Find sin⁡θ.
  4. The point (−3,−4) is on the terminal side of θ. Find tan⁡θ.
  5. The point (24,−7) is on the terminal side of θ. Find tan⁡θ.
  6. The point (5,12) is on the terminal side of θ. Find cos⁡θ.
  7. The point (−11,−60) is on the terminal side of θ. Find sin⁡θ.
  8. The point (−15,8) is on the terminal side of θ. Find sin⁡θ.
  9. The point (−11,60) is on the terminal side of θ. Find cos⁡θ.
  10. The point (56,−33) is on the terminal side of θ. Find cos⁡θ.

  1. Find several ratios from a point

    The point (3,−5) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.
  2. The point (1,3) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.
  3. The point (−1,1) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.
  4. The point (−3,−6) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.
  5. The point (6,−2) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.
  6. The point (2,5) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.
  7. The point (−4,5) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.
  8. The point (−4,−1) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.
  9. The point (−3,2) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.
  10. The point (6,−5) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Trig Ratios of General AnglesAnswer keyVersion 2

Date Period

Find each ratio.

  1. Find a ratio from a point

    The point (60,11) is on the terminal side of θ. Find tan⁡θ.1160
  2. The point (−5,12) is on the terminal side of θ. Find tan⁡θ.−125
  3. The point (48,−55) is on the terminal side of θ. Find sin⁡θ.−5573
  4. The point (−3,−4) is on the terminal side of θ. Find tan⁡θ.43
  5. The point (24,−7) is on the terminal side of θ. Find tan⁡θ.−724
  6. The point (5,12) is on the terminal side of θ. Find cos⁡θ.513
  7. The point (−11,−60) is on the terminal side of θ. Find sin⁡θ.−6061
  8. The point (−15,8) is on the terminal side of θ. Find sin⁡θ.817
  9. The point (−11,60) is on the terminal side of θ. Find cos⁡θ.−1161
  10. The point (56,−33) is on the terminal side of θ. Find cos⁡θ.5665

  1. Find several ratios from a point

    The point (3,−5) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.sin⁡θ=−53434, cos⁡θ=33434, tan⁡θ=−53, csc⁡θ=−345, sec⁡θ=343, cot⁡θ=−35
  2. The point (1,3) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.sin⁡θ=31010, cos⁡θ=1010, tan⁡θ=3, csc⁡θ=103, sec⁡θ=10, cot⁡θ=13
  3. The point (−1,1) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.sin⁡θ=22, cos⁡θ=−22, tan⁡θ=−1, csc⁡θ=2, sec⁡θ=−2, cot⁡θ=−1
  4. The point (−3,−6) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.sin⁡θ=−255, cos⁡θ=−55, tan⁡θ=2, csc⁡θ=−52, sec⁡θ=−5, cot⁡θ=12
  5. The point (6,−2) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.sin⁡θ=−1010, cos⁡θ=31010, tan⁡θ=−13, csc⁡θ=−10, sec⁡θ=103, cot⁡θ=−3
  6. The point (2,5) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.sin⁡θ=52929, cos⁡θ=22929, tan⁡θ=52, csc⁡θ=295, sec⁡θ=292, cot⁡θ=25
  7. The point (−4,5) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.sin⁡θ=54141, cos⁡θ=−44141, tan⁡θ=−54, csc⁡θ=415, sec⁡θ=−414, cot⁡θ=−45
  8. The point (−4,−1) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.sin⁡θ=−1717, cos⁡θ=−41717, tan⁡θ=14, csc⁡θ=−17, sec⁡θ=−174, cot⁡θ=4
  9. The point (−3,2) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.sin⁡θ=21313, cos⁡θ=−31313, tan⁡θ=−23, csc⁡θ=132, sec⁡θ=−133, cot⁡θ=−32
  10. The point (6,−5) is on the terminal side of θ. Find sin⁡θ, cos⁡θ, tan⁡θ, csc⁡θ, sec⁡θ, cot⁡θ.sin⁡θ=−56161, cos⁡θ=66161, tan⁡θ=−56, csc⁡θ=−615, sec⁡θ=616, cot⁡θ=−65

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

How to do these

The point (−5,−12) is on the terminal side of θ. Find cos⁡θ

  1. r = √((-5)² + (-12)²) = √169 = 13.
  2. Cosine is x over r.

The answer is −513.

Where students go wrong

Dropping the sign of a coordinate. r is always positive, but x and y keep their signs, so for (-5, -12) the cosine is -5/13, not 5/13.

Also called trig functions of any angle, terminal side or ASTC.

What you can put on this worksheet

A ratio from a point on the terminal side
The point (−5,−12) is on the terminal side of θ. Find cos⁡θ. → −513
One ratio and the quadrant, find another
If sin⁡θ=−1517 and θ is in quadrant III, find cos⁡θ → −817
Every ratio from a point, with radicals
The point (4,1) is on the terminal side of θ. Find sin⁡θ,cos⁡θ. → sin⁡θ=1717, cos⁡θ=41717
Calculator values, drawn points, points with roots
Find sin⁡θ,cos⁡θ,tan⁡θ — an angle in standard position, labeled x, y → sin⁡θ=−21313 cos⁡θ=31313 tan⁡θ=−23
One ratio and a sign: the others, with radicals
sin⁡θ=41717 and cos⁡θ>0. Find the other five ratios. → cos⁡θ=1717tan⁡θ=4csc⁡θ=174sec⁡θ=17cot⁡θ=14

Questions about these worksheets

Yes. Every trig ratios of general angles 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 trig ratios of general angles 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 5: a ratio from a point on the terminal side, one ratio and the quadrant, find another, every ratio from a point, with radicals, calculator values, drawn points, points with roots and one ratio and a sign: the others, with radicals. Tick as many as you want and set how many of each, or let it spread them evenly.

Dropping the signs. The radius is always positive, but x and y carry their own signs, so a point in the second quadrant gives a positive sine and a negative cosine. Sketching the point takes five seconds and settles it.