Write a trigonometric ratioWrite all six trigonometric ratios
Name: Date:
Write each ratio as a fraction. Read the sides straight off the triangle.
1 sin B =
2 sec B =
3 cot A =
4 csc A =
5 tan B =
6 cot A =
7 cos B =
8 csc A =
- Free
- Grade 9
- Pythagorean triples, six ratios
The six trig ratios worksheets
Sine, cosine and tangent each have a reciprocal — cosecant, secant and cotangent — so a right triangle has six ratios, not three. These free printable worksheets give a right triangle with whole-number sides and ask for all six as fractions.
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Ready to printThe six trig ratios
The answer key prints on a separate page.
What's on this sheet
The three new ratios are just the first three flipped: cosecant is hypotenuse over opposite, secant is hypotenuse over adjacent, cotangent is adjacent over opposite. Every triangle here is a Pythagorean triple — 3-4-5, 5-12-13, 8-15-17 and the rest — so all three sides are whole numbers and each of the six ratios reduces to a clean fraction. That is what lets the page skip the calculator: csc A is the opposite flipped under the hypotenuse, read straight off the figure and written as it stands.
Why this sheet works
The skill is naming the sides from the marked angle and then keeping the pairs straight — which is over which. Sine and cosecant share the same two sides in opposite order, and so do the other two pairs, so a student who can write sin A can write csc A by turning the fraction over. Reading the same triangle for all six is what fixes the reciprocal relationship in place.
How this one works
One question from this sheet, worked through a step at a time.
Write sin B as a fraction
sin B=opposite / hypotenuse
sin B=21/29
sin B=21/29
- sin B is opposite over hypotenuse. Name the two sides you need, measured from angle B.
- Read them off the triangle: the opposite side of angle B is 21, the hypotenuse is 29.
- So sin B = 21/29.