Trigonometry worksheet for calculating sine, cosine, and tangent values of standard angles.

Let’s solve this worksheet step by step. We’ll go through each part carefully.

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Part 1: Find sin, cos, and tan ratios for each right triangle.

We are given three triangles (A, B, C). For each, we need to find the trigonometric ratios for the marked angles.

Remember:
- sin = opposite / hypotenuse
- cos = adjacent / hypotenuse
- tan = opposite / adjacent

Also, simplify fractions and round decimals to 2 places.

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Triangle A:
Right triangle with sides: opposite = 3, adjacent = 4, hypotenuse = 5
Angle P is at the bottom left — so for angle P:
- Opposite = 3
- Adjacent = 4
- Hypotenuse = 5

So:
1. sin P = 3/5 = 0.60
2. cos P = 4/5 = 0.80
3. tan P = 3/4 = 0.75

Angle Q is at the top — so for angle Q:
- Opposite = 4
- Adjacent = 3
- Hypotenuse = 5

So:
4. sin Q = 4/5 = 0.80
5. cos Q = 3/5 = 0.60
6. tan Q = 4/3 ≈ 1.33 (since 4 ÷ 3 = 1.333...)

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Triangle B:
Sides: opposite = 7, adjacent = 24, hypotenuse = 25
Angle X is at the bottom left — so for angle X:
- Opposite = 7
- Adjacent = 24
- Hypotenuse = 25

So:
7. sin X = 7/25 = 0.28
8. cos X = 24/25 = 0.96
9. tan X = 7/24 ≈ 0.29 (7 ÷ 24 = 0.2916... → 0.29)

Angle Y is at the top — so for angle Y:
- Opposite = 24
- Adjacent = 7
- Hypotenuse = 25

So:
10. sin Y = 24/25 = 0.96
11. cos Y = 7/25 = 0.28
12. tan Y = 24/7 ≈ 3.43 (24 ÷ 7 = 3.428... → 3.43)

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Triangle C:
This one has sides labeled: 8, 15, 17 (it’s a right triangle because 8² + 15² = 64 + 225 = 289 = 17²)
Angle X is at the bottom left — so for angle X:
- Opposite = 8
- Adjacent = 15
- Hypotenuse = 17

So:
13. sin X = 8/17 ≈ 0.47 (8 ÷ 17 = 0.4705... → 0.47)
14. cos X = 15/17 ≈ 0.88 (15 ÷ 17 = 0.8823... → 0.88)
15. tan X = 8/15 ≈ 0.53 (8 ÷ 15 = 0.5333... → 0.53)

Angle Y is at the top — so for angle Y:
- Opposite = 15
- Adjacent = 8
- Hypotenuse = 17

So:
16. sin Y = 15/17 ≈ 0.88
17. cos Y = 8/17 ≈ 0.47
18. tan Y = 15/8 = 1.875 → 1.88 (rounded to 2 decimal places)

Wait — let me double-check that last one: 15 ÷ 8 = 1.875 → yes, rounds to 1.88.

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Part 2: Use calculator to find decimal values of sine, cosine, tangent for given angles.

We’ll use a calculator (make sure it’s in degree mode!).

Round to 2 decimal places unless specified otherwise.

19. sin 23° → 0.39
20. cos 39° → 0.78
21. sin 59° → 0.86
22. sin 48° → 0.74
23. sin 36° → 0.59
24. sin 58° → 0.85
25. cos 10° → 0.98
26. cos 28° → 0.88
27. cos 50° → 0.64
28. cos 40° → 0.77
29. sin 50° → 0.77
30. cos 60° → 0.50
31. tan 10° → 0.18
32. tan 35° → 0.70
33. tan 59° → 1.66
34. tan 40° → 0.84
35. tan 48° → 1.11
36. tan 60° → 1.73
37. tan 70° → 2.75
38. tan 89° → 57.29 (very large, since tan approaches infinity as angle nears 90°)
39. tan 89.5° → 114.59
40. tan 89.9° → 572.96
41. tan 89.99° → 5729.58

Note: As the angle gets closer to 90°, tangent becomes very large — which makes sense because tan = sin/cos, and cos gets close to 0.

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Part 3: Think About It!

Question: Why are sine and cosine ratios always less than 1, but tangents are not?

Answer:

In a right triangle:
- Sine = opposite / hypotenuse → opposite side is ALWAYS shorter than hypotenuse → so ratio < 1
- Cosine = adjacent / hypotenuse → adjacent side is ALSO always shorter than hypotenuse → so ratio < 1
- Tangent = opposite / adjacent → either side can be longer than the other → so tangent can be less than 1, equal to 1, or greater than 1!

Example: In Triangle A, tan Q = 4/3 > 1. In Triangle B, tan X = 7/24 < 1.

So tangent isn’t limited like sine and cosine.

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Final Answer:

Here are all the answers filled in order:

Part 1:

1. sin P = 3/5 = 0.60
2. cos P = 4/5 = 0.80
3. tan P = 3/4 = 0.75
4. sin Q = 4/5 = 0.80
5. cos Q = 3/5 = 0.60
6. tan Q = 4/3 ≈ 1.33
7. sin X = 7/25 = 0.28
8. cos X = 24/25 = 0.96
9. tan X = 7/24 ≈ 0.29
10. sin Y = 24/25 = 0.96
11. cos Y = 7/25 = 0.28
12. tan Y = 24/7 ≈ 3.43
13. sin X = 8/17 ≈ 0.47
14. cos X = 15/17 ≈ 0.88
15. tan X = 8/15 ≈ 0.53
16. sin Y = 15/17 ≈ 0.88
17. cos Y = 8/17 ≈ 0.47
18. tan Y = 15/8 = 1.88

Part 2:

19. 0.39
20. 0.78
21. 0.86
22. 0.74
23. 0.59
24. 0.85
25. 0.98
26. 0.88
27. 0.64
28. 0.77
29. 0.77
30. 0.50
31. 0.18
32. 0.70
33. 1.66
34. 0.84
35. 1.11
36. 1.73
37. 2.75
38. 57.29
39. 114.59
40. 572.96
41. 5729.58

Part 3:

Sine and cosine are always less than 1 because they compare a leg to the hypotenuse — and the hypotenuse is always the longest side. Tangent compares two legs — one can be longer than the other, so tangent can be greater than 1.