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Trigonometric Ratios - Worksheet 8 - 10th Grade

Hard

Define and apply sine, cosine, and tangent in right triangles (Hard)

30 problems · 20-25 minutes · Skills: trigonometry, right triangles, ratios

About This Worksheet

This hard-level worksheet contains 30 problems designed for 10th Grade students practicing trigonometric ratios. Hard worksheets feature challenging problems including multi-step questions, missing values, and real-world applications to push students further.

Trigonometric ratios are one of the most powerful tools in all of mathematics, connecting angles to lengths in a way that enables calculation of distances, heights, and positions that cannot be measured directly. They are the gateway to a vast portion of mathematics — from the unit circle to Fourier analysis — and have applications in every field of science and engineering.

  • Define sine, cosine, and tangent as ratios of sides in a right triangle using SOH-CAH-TOA
  • Write the six trigonometric ratios for an acute angle given a right triangle
  • Use a calculator to evaluate trig ratios and inverse trig functions for any acute angle
  • Use reciprocal ratios (cosecant, secant, cotangent) to solve for missing sides
  • Identify co-function relationships between complementary angles

How to Solve: Worked Example

Example Problem: A right triangle has legs 8 and 15. Find the hypotenuse, then sin of the angle opposite 8.
Solution Steps:
  1. Hypotenuse: √(8² + 15²) = √(64 + 225) = √289 = 17.
  2. The side opposite the angle is 8.
  3. sin = opposite/hypotenuse = 8/17.
Answer: hypotenuse 17, sin = 8/17
Tip: Find the missing side before applying any ratio — you cannot use SOH-CAH-TOA without all the pieces you need.
1
A right triangle has legs 5 and 12. Find the hypotenuse, then sin of the angle opposite the side of length 5.
2
A right triangle has legs 12 and 35. Find the hypotenuse, then sin of the angle opposite the side of length 12.
3
A right triangle has legs 6 and 8. Find the hypotenuse, then sin of the angle opposite the side of length 6.
4
A right triangle has legs 30 and 16. Find the hypotenuse, then sin of the angle opposite the side of length 30.
5
A right triangle has legs 24 and 18. Find the hypotenuse, then sin of the angle opposite the side of length 24.
6
A right triangle has legs 15 and 8. Find the hypotenuse, then sin of the angle opposite the side of length 15.
7
A right triangle has legs 3 and 4. Find the hypotenuse, then sin of the angle opposite the side of length 3.
8
A right triangle has legs 14 and 48. Find the hypotenuse, then sin of the angle opposite the side of length 14.
9
A right triangle has legs 21 and 20. Find the hypotenuse, then sin of the angle opposite the side of length 21.
10
A right triangle has legs 48 and 55. Find the hypotenuse, then sin of the angle opposite the side of length 48.
11
A right triangle has legs 4 and 3. Find the hypotenuse, then sin of the angle opposite the side of length 4.
12
A right triangle has legs 15 and 8. Find the hypotenuse, then sin of the angle opposite the side of length 15.
13
A right triangle has legs 21 and 20. Find the hypotenuse, then sin of the angle opposite the side of length 21.
14
A right triangle has legs 5 and 12. Find the hypotenuse, then sin of the angle opposite the side of length 5.
15
A right triangle has legs 24 and 10. Find the hypotenuse, then sin of the angle opposite the side of length 24.
16
A right triangle has legs 36 and 15. Find the hypotenuse, then sin of the angle opposite the side of length 36.
17
A right triangle has legs 10 and 24. Find the hypotenuse, then sin of the angle opposite the side of length 10.
18
A right triangle has legs 18 and 24. Find the hypotenuse, then sin of the angle opposite the side of length 18.
19
A right triangle has legs 45 and 28. Find the hypotenuse, then sin of the angle opposite the side of length 45.
20
A right triangle has legs 36 and 15. Find the hypotenuse, then sin of the angle opposite the side of length 36.
21
A right triangle has legs 24 and 10. Find the hypotenuse, then sin of the angle opposite the side of length 24.
22
A right triangle has legs 20 and 15. Find the hypotenuse, then sin of the angle opposite the side of length 20.
23
A right triangle has legs 20 and 15. Find the hypotenuse, then sin of the angle opposite the side of length 20.
24
A right triangle has legs 5 and 12. Find the hypotenuse, then sin of the angle opposite the side of length 5.
25
A right triangle has legs 30 and 16. Find the hypotenuse, then sin of the angle opposite the side of length 30.
26
A right triangle has legs 63 and 16. Find the hypotenuse, then sin of the angle opposite the side of length 63.
27
A right triangle has legs 12 and 9. Find the hypotenuse, then sin of the angle opposite the side of length 12.
28
A right triangle has legs 60 and 11. Find the hypotenuse, then sin of the angle opposite the side of length 60.
29
A right triangle has legs 24 and 7. Find the hypotenuse, then sin of the angle opposite the side of length 24.
30
A right triangle has legs 12 and 35. Find the hypotenuse, then sin of the angle opposite the side of length 12.

Answer Key

1. hypotenuse 13, sin = 5/132. hypotenuse 37, sin = 12/373. hypotenuse 10, sin = 3/54. hypotenuse 34, sin = 15/175. hypotenuse 30, sin = 4/56. hypotenuse 17, sin = 15/177. hypotenuse 5, sin = 3/58. hypotenuse 50, sin = 7/259. hypotenuse 29, sin = 21/2910. hypotenuse 73, sin = 48/7311. hypotenuse 5, sin = 4/512. hypotenuse 17, sin = 15/1713. hypotenuse 29, sin = 21/2914. hypotenuse 13, sin = 5/1315. hypotenuse 26, sin = 12/1316. hypotenuse 39, sin = 12/1317. hypotenuse 26, sin = 5/1318. hypotenuse 30, sin = 3/519. hypotenuse 53, sin = 45/5320. hypotenuse 39, sin = 12/1321. hypotenuse 26, sin = 12/1322. hypotenuse 25, sin = 4/523. hypotenuse 25, sin = 4/524. hypotenuse 13, sin = 5/1325. hypotenuse 34, sin = 15/1726. hypotenuse 65, sin = 63/6527. hypotenuse 15, sin = 4/528. hypotenuse 61, sin = 60/6129. hypotenuse 25, sin = 24/2530. hypotenuse 37, sin = 12/37

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What skills does this right triangle trigonometry worksheet cover?

This worksheet focuses on trigonometry, right triangles, ratios. It is designed for 10th Grade students and provides structured practice through varied problem types with a worked example to guide students before they begin.

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How many problems are on this worksheet?

This worksheet contains 30 problems. It typically takes 20-25 minutes to complete. An answer key with all solutions is included so parents and teachers can check work quickly.