Trigonometric Ratios Worksheets - Math Monks - Free Printable

Let’s solve each problem one by one. We’ll use the basic trigonometric ratios for right triangles:

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

We always look at the angle given, then find which side is opposite, which is adjacent, and which is the hypotenuse (the longest side, across from the right angle).

---

Problem 1: tan A

Triangle ABC, right angle at B.

Angle A →
Opposite side = BC = 21
Adjacent side = AB = 20
→ tan A = opposite/adjacent = 21/20

✔ Check: Yes, angle A is at top left, so side opposite is bottom (BC=21), adjacent is vertical (AB=20). Correct.

---

Problem 2: cos C

Triangle ABC, right angle at B.

Angle C →
Adjacent side = BC = 16
Hypotenuse = AC = 34
→ cos C = adjacent/hypotenuse = 16/34 = simplify? Let’s see: divide numerator and denominator by 2 → 8/17

But unless asked to simplify, we can leave as 16/34. But usually simplified is better. Let’s go with 8/17.

Wait — let me double-check: Angle C is at bottom left. Adjacent side is next to it along the base: BC = 16. Hypotenuse is AC = 34. Yes.

So cos C = 16/34 = 8/17

---

Problem 3: sin Z

Triangle XYZ, right angle at Y.

Angle Z →
Opposite side = XY = 12
Hypotenuse = XZ = 37
→ sin Z = opposite/hypotenuse = 12/37

Check: Angle Z is at left end. Opposite side is the one not touching Z except at vertex — that’s XY = 12. Hypotenuse is XZ = 37. Correct.

---

Problem 4: tan X

Triangle XYZ, right angle at Y.

Angle X →
Opposite side = YZ = 30
Adjacent side = XY = 40
→ tan X = 30/40 = 3/4

Check: Angle X is at bottom left. Opposite is up to YZ = 30. Adjacent is along XY = 40. Yes.

---

Problem 5: cos X

Triangle XYZ, right angle at Z.

Angle X →
Adjacent side = XZ = 2√2
Hypotenuse = XY = 3
→ cos X = adjacent/hypotenuse = (2√2)/3

No simplification needed. Leave as is.

Check: Right angle at Z. So sides: XZ and YZ are legs, XY is hypotenuse. Angle X: adjacent is XZ = 2√2, yes.

---

Problem 6: tan A

Triangle ABC, right angle at C.

Angle A →
Opposite side = BC = 4
Adjacent side = AC = 3
→ tan A = 4/3

Check: Angle A is at top left. Opposite is down to BC = 4. Adjacent is right to AC = 3. Yes.

---

Problem 7: cos X

Triangle XYZ, right angle at Y.

Angle X →
Adjacent side = XY = 21
Hypotenuse = XZ = 29
→ cos X = 21/29

Check: Angle X is at bottom left. Adjacent is along XY = 21. Hypotenuse is XZ = 29. Yes.

---

Problem 8: sin C

Triangle ABC, right angle at B.

Angle C →
Opposite side = AB = 14
Hypotenuse = AC = 50
→ sin C = 14/50 = 7/25 (simplified)

Check: Angle C is at right end. Opposite side is AB = 14 (across from C). Hypotenuse is AC = 50. Yes.

Simplify 14/50 → divide by 2 → 7/25.

---

Now, compiling all answers:

1. tan A = 21/20
2. cos C = 8/17
3. sin Z = 12/37
4. tan X = 3/4
5. cos X = (2√2)/3
6. tan A = 4/3
7. cos X = 21/29
8. sin C = 7/25

All checked carefully.

Final Answer:
1. \frac{21}{20}
2. \frac{8}{17}
3. \frac{12}{37}
4. \frac{3}{4}
5. \frac{2\sqrt{2}}{3}
6. \frac{4}{3}
7. \frac{21}{29}
8. \frac{7}{25}