Trigonometic Ratios (SOH CAH TOA): Word Problems (With Clues ... - Free Printable

Let’s solve each problem one by one, step by step. We’ll use trigonometry (sine, cosine, tangent) and basic geometry where needed. Remember: in right triangles, we can use SOH CAH TOA:

- Sin = Opposite / Hypotenuse
- Cos = Adjacent / Hypotenuse
- Tan = Opposite / Adjacent

Also, angles in a triangle add up to 180°, and in a right triangle, the two non-right angles add to 90°.

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Problem 1:

ABC is a right triangle. AB = 7 cm, angle ABC = 90°, angle ACB = 64°. Find BC.

We have:
- Right angle at B → so sides AB and BC are legs, AC is hypotenuse.
- Angle at C is 64° → so from point C, side AB is opposite, and BC is adjacent.
- So for angle C (64°), we know:
- Opposite = AB = 7 cm
- Adjacent = BC = ? (what we want)
- Use tan(θ) = opposite / adjacent

So:
tan(64°) = 7 / BC
→ BC = 7 / tan(64°)

Calculate tan(64°):
Using calculator: tan(64°) ≈ 2.0503

Then:
BC = 7 / 2.0503 ≈ 3.414

To 3 significant figures: 3.41 cm

✔ Check: Makes sense — since angle at C is large (64°), the adjacent side should be shorter than opposite? Wait — no! Actually, if angle at C is 64°, then angle at A is 26°, so side opposite 64° (which is AB=7) should be longer than side opposite 26° (which is BC). Yes — 7 > 3.41 → correct.

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Problem 2:

Right triangle with sides 5 cm, 12 cm, 13 cm. Find the other two angles.

First, confirm it’s a right triangle: 5² + 12² = 25 + 144 = 169 = 13² → yes, right-angled between 5 and 12.

So, let’s say:
- Side opposite angle A = 5
- Side opposite angle B = 12
- Hypotenuse = 13

We can find angle opposite 5 cm side using sin or tan.

Use sin(θ) = opposite/hypotenuse = 5/13 ≈ 0.3846
→ θ = arcsin(0.3846) ≈ 22.6°

Other angle = 90° - 22.6° = 67.4°

Check with tan: tan(θ) = 5/12 ≈ 0.4167 → arctan(0.4167) ≈ 22.6° → same.

So angles are approximately 22.6° and 67.4°

To 3 sig figs: 22.6° and 67.4°

✔ Check: 22.6 + 67.4 = 90 → good.

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Problem 3:

Perimeter of right triangle = 24 cm. One side = 10 cm, another = 8 cm. Find smallest angle.

First, find third side: perimeter = sum of all sides → 24 = 10 + 8 + x → x = 6 cm

So sides are 6, 8, 10 → check if right triangle: 6² + 8² = 36 + 64 = 100 = 10² → yes! Right-angled between 6 and 8.

Smallest angle is opposite the shortest side → opposite 6 cm.

So, sin(θ) = opposite/hypotenuse = 6/10 = 0.6
→ θ = arcsin(0.6) ≈ 36.87°

To 3 sig figs: 36.9°

✔ Check: Other angle = 90 - 36.9 = 53.1° → larger → correct.

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Problem 4:

Rectangle: length 14 cm, width 8 cm. What angle does diagonal make with longest side?

Longest side is 14 cm (length). Diagonal forms a right triangle with sides 14 and 8.

Angle with longest side → that’s the angle between diagonal and 14 cm side.

In that right triangle:
- Adjacent to angle = 14 cm
- Opposite = 8 cm
- So tan(θ) = opposite/adjacent = 8/14 ≈ 0.5714

θ = arctan(0.5714) ≈ 29.7°

To 3 sig figs: 29.7°

✔ Check: Makes sense — smaller angle with longer side.

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Problem 5:

Ladder 5 m long, makes 82° with ground. How far up wall?

This is a right triangle:
- Hypotenuse = ladder = 5 m
- Angle with ground = 82°
- Height up wall = opposite side to 82°

So sin(82°) = opposite / hypotenuse = height / 5
→ height = 5 × sin(82°)

sin(82°) ≈ 0.9903
→ height = 5 × 0.9903 ≈ 4.9515 m

To 3 sig figs: 4.95 m

✔ Check: Since angle is very steep (82°), height should be almost full length → 4.95 is close to 5 → makes sense.

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Final Answer:
1) 3.41 cm
2) 22.6° and 67.4°
3) 36.9°
4) 29.7°
5) 4.95 m