Comprehensive trigonometry worksheet designed to help students practice calculating the lengths of unknown sides in right-angled triangles.

Let’s solve each part of the worksheet step by step, explaining the trigonometry concepts involved.

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## Notes Section Recap (for reference):

In a right triangle:
- o = opposite side to the given angle
- a = adjacent side to the given angle
- h = hypotenuse (side opposite the right angle)

The three main trig ratios are:
- sin(θ) = o / h
- cos(θ) = a / h
- tan(θ) = o / a

You choose the ratio based on which sides you know and which you need to find.

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# Exercise 1: Calculate the unknown side in each equation

These are straightforward algebraic rearrangements using the given trig equations.

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a) sin 20° = a / 15

We want to solve for a.

Multiply both sides by 15:

> a = 15 × sin 20°

Use calculator (in degree mode):

> sin 20° ≈ 0.3420
> a ≈ 15 × 0.3420 = 5.13

✔ Answer: a ≈ 5.13

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b) cos 37° = b / 30

Solve for b:

> b = 30 × cos 37°

cos 37° ≈ 0.7986

> b ≈ 30 × 0.7986 = 23.96

✔ Answer: b ≈ 23.96

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c) tan 64° = c / 5

Solve for c:

> c = 5 × tan 64°

tan 64° ≈ 2.0503

> c ≈ 5 × 2.0503 = 10.25

✔ Answer: c ≈ 10.25

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d) sin 52° = d / 32

Solve for d:

> d = 32 × sin 52°

sin 52° ≈ 0.7880

> d ≈ 32 × 0.7880 = 25.22

✔ Answer: d ≈ 25.22

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e) cos 10° = e / 18

Solve for e:

> e = 18 × cos 10°

cos 10° ≈ 0.9848

> e ≈ 18 × 0.9848 = 17.73

✔ Answer: e ≈ 17.73

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f) tan 25° = f / 22

Solve for f:

> f = 22 × tan 25°

tan 25° ≈ 0.4663

> f ≈ 22 × 0.4663 = 10.26

✔ Answer: f ≈ 10.26

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# Exercise 2: Calculate lengths of lettered sides (all in meters)

Each diagram is a right triangle with one angle and one side given. We use SOH-CAH-TOA to pick the correct ratio.

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Top Left Triangle: Angle = 60°, hypotenuse = 12, find ‘a’ (adjacent)

Since we have angle and hypotenuse, and want adjacent, use cos:

> cos(60°) = adjacent / hypotenuse = a / 12
> → a = 12 × cos(60°)

cos(60°) = 0.5

> a = 12 × 0.5 = 6 m

✔ Answer: a = 6

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Top Middle Triangle: Angle = 45°, hypotenuse = 8, find ‘c’ (opposite)

We have angle and hypotenuse, want opposite → use sin

> sin(45°) = opposite / hypotenuse = c / 8
> → c = 8 × sin(45°)

sin(45°) = √2/2 ≈ 0.7071

> c ≈ 8 × 0.7071 = 5.66 m

✔ Answer: c ≈ 5.66

*(Note: In a 45-45-90 triangle, legs are equal and = hypotenuse / √2 → 8/√2 = 4√2 ≈ 5.66 — same result)*

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Top Right Triangle: Angle = 25°, hypotenuse = 15, find ‘e’ (adjacent)

Angle and hypotenuse → want adjacent → use cos

> cos(25°) = e / 15
> → e = 15 × cos(25°)

cos(25°) ≈ 0.9063

> e ≈ 15 × 0.9063 = 13.59 m

✔ Answer: e ≈ 13.59

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Bottom Left Triangle: Angle = 35°, adjacent = 24, find ‘b’ (opposite)

We have angle and adjacent, want opposite → use tan

> tan(35°) = opposite / adjacent = b / 24
> → b = 24 × tan(35°)

tan(35°) ≈ 0.7002

> b ≈ 24 × 0.7002 = 16.80 m

✔ Answer: b ≈ 16.80

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Bottom Middle Triangle: Angle = 26°, hypotenuse = 15, find ‘d’ (adjacent)

Angle and hypotenuse → want adjacent → use cos

> cos(26°) = d / 15
> → d = 15 × cos(26°)

cos(26°) ≈ 0.8988

> d ≈ 15 × 0.8988 = 13.48 m

✔ Answer: d ≈ 13.48

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Bottom Right Triangle: Angle = 70°, hypotenuse = 17, find ‘f’ (opposite)

Angle and hypotenuse → want opposite → use sin

> sin(70°) = f / 17
> → f = 17 × sin(70°)

sin(70°) ≈ 0.9397

> f ≈ 17 × 0.9397 = 15.97 m

✔ Answer: f ≈ 15.97

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# Exercise 3: Flagpole Problem

> A flagpole is held upright by 30m long wires to the ground. If the wires make an angle of 25° to the ground, how high is the flagpole?

This is a classic right triangle problem.

- The wire is the hypotenuse = 30m
- The angle with the ground = 25° → so this is the angle between the wire and the ground
- The height of the flagpole is the opposite side to the 25° angle

So we use sin:

> sin(25°) = opposite / hypotenuse = height / 30
> → height = 30 × sin(25°)

sin(25°) ≈ 0.4226

> height ≈ 30 × 0.4226 = 12.68 m

✔ Answer: The flagpole is approximately 12.68 meters high.

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## ✔ Final Answers Summary:

Exercise 1:

a) 5.13
b) 23.96
c) 10.25
d) 25.22
e) 17.73
f) 10.26

Exercise 2:

a) 6 m
b) 16.80 m
c) 5.66 m
d) 13.48 m
e) 13.59 m
f) 15.97 m

Exercise 3:

Flagpole height = 12.68 m

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## 📌 Key Takeaway:

Always identify:
- Which side you’re given (opposite, adjacent, hypotenuse)
- Which side you’re solving for
- Which angle you’re using

Then choose the correct trig ratio (SOH-CAH-TOA) and solve algebraically.

Let me know if you’d like diagrams or visual explanations too!