Law of Sines practice worksheet with eight triangle problems to find missing sides and angles.

Let’s solve each problem one by one using the Law of Sines.

The Law of Sines says:

> In any triangle,
> \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)
> where \(a, b, c\) are side lengths and \(A, B, C\) are the angles opposite those sides.

We’ll use this to find missing sides or angles. Remember: the sum of angles in a triangle is always 180° — we’ll use that when needed.

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Problem 1:

Triangle with angles 53° and 40°, side opposite 40° is 5, find side x opposite 53°.

First, find the third angle:
180° - 53° - 40° = 87°

Now apply Law of Sines:

\(\frac{x}{\sin 53^\circ} = \frac{5}{\sin 40^\circ}\)

Solve for x:

\(x = \frac{5 \cdot \sin 53^\circ}{\sin 40^\circ}\)

Use calculator (make sure it’s in degree mode):

sin 53° ≈ 0.7986
sin 40° ≈ 0.6428

So,

\(x ≈ \frac{5 \cdot 0.7986}{0.6428} ≈ \frac{3.993}{0.6428} ≈ 6.21\)

✔ Answer: x ≈ 6.21

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Problem 2:

Angles given: 128° and 32°, side opposite 32° is 10, find side x opposite 128°.

Wait — let’s check: if two angles are 128° and 32°, then third angle is:

180 - 128 - 32 = 20°

But looking at the diagram: side labeled “10” is opposite the 32° angle? Actually, from the drawing, side “10” is between the 128° and 32° angles — so it’s opposite the *third* angle, which is 20°.

Actually, let me re-read the diagram description.

In problem 2: triangle has angles 128° and 32°, and side labeled “10” is adjacent to both? No — standard labeling: side opposite an angle is across from it.

Looking again: the side labeled “10” is opposite the angle that’s NOT labeled — which must be 20°.

And side “x” is opposite the 128° angle.

So:

\(\frac{x}{\sin 128^\circ} = \frac{10}{\sin 20^\circ}\)

Compute:

sin 128° = sin(180-52) = sin 52° ≈ 0.7880
sin 20° ≈ 0.3420

So:

\(x = \frac{10 \cdot 0.7880}{0.3420} ≈ \frac{7.88}{0.3420} ≈ 23.04\)

✔ Answer: x ≈ 23.04

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Problem 3:

Angles: 99° and 50°, side opposite 50° is 9, find side x opposite 99°.

Third angle: 180 - 99 - 50 = 31°

Law of Sines:

\(\frac{x}{\sin 99^\circ} = \frac{9}{\sin 50^\circ}\)

sin 99° ≈ 0.9877
sin 50° ≈ 0.7660

\(x = \frac{9 \cdot 0.9877}{0.7660} ≈ \frac{8.8893}{0.7660} ≈ 11.60\)

✔ Answer: x ≈ 11.60

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Problem 4:

Angles: 62° and 65°, side opposite 65° is 6, find side x opposite 62°.

Third angle: 180 - 62 - 65 = 53°

Law of Sines:

\(\frac{x}{\sin 62^\circ} = \frac{6}{\sin 65^\circ}\)

sin 62° ≈ 0.8829
sin 65° ≈ 0.9063

\(x = \frac{6 \cdot 0.8829}{0.9063} ≈ \frac{5.2974}{0.9063} ≈ 5.84\)

✔ Answer: x ≈ 5.84

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Problem 5:

Angles: 110° and 46°, side opposite 46° is 15, find side x opposite 110°.

Third angle: 180 - 110 - 46 = 24°

Law of Sines:

\(\frac{x}{\sin 110^\circ} = \frac{15}{\sin 46^\circ}\)

sin 110° = sin(70°) ≈ 0.9397
sin 46° ≈ 0.7193

\(x = \frac{15 \cdot 0.9397}{0.7193} ≈ \frac{14.0955}{0.7193} ≈ 19.59\)

✔ Answer: x ≈ 19.59

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Problem 6:

Angles: 81° and 61°, side opposite 81° is 7, find side x opposite 61°.

Third angle: 180 - 81 - 61 = 38°

Law of Sines:

\(\frac{x}{\sin 61^\circ} = \frac{7}{\sin 81^\circ}\)

sin 61° ≈ 0.8746
sin 81° ≈ 0.9877

\(x = \frac{7 \cdot 0.8746}{0.9877} ≈ \frac{6.1222}{0.9877} ≈ 6.20\)

✔ Answer: x ≈ 6.20

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Problem 7:

Find all missing sides and angles.

Given: one side = 12, angles 41° and 76°.

First, find third angle: 180 - 41 - 76 = 63°

Label triangle:

Let’s say:
- Angle A = 41°, opposite side a = ?
- Angle B = 76°, opposite side b = ?
- Angle C = 63°, opposite side c = 12

Use Law of Sines:

\(\frac{a}{\sin 41^\circ} = \frac{b}{\sin 76^\circ} = \frac{12}{\sin 63^\circ}\)

First, compute common ratio:

sin 63° ≈ 0.8910 → 12 / 0.8910 ≈ 13.468

Now:

Side a (opposite 41°):
a = 13.468 × sin 41° ≈ 13.468 × 0.6561 ≈ 8.84

Side b (opposite 76°):
b = 13.468 × sin 76° ≈ 13.468 × 0.9703 ≈ 13.07

✔ Answers:
- Missing angle: 63°
- Side opposite 41°: ≈ 8.84
- Side opposite 76°: ≈ 13.07

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Problem 8:

Find all missing sides and angles.

Given: angles 95° and 48°, side opposite 48° is 16.

Third angle: 180 - 95 - 48 = 37°

Label:

- Angle A = 95°, opposite side a = ?
- Angle B = 48°, opposite side b = 16
- Angle C = 37°, opposite side c = ?

Law of Sines:

\(\frac{a}{\sin 95^\circ} = \frac{16}{\sin 48^\circ} = \frac{c}{\sin 37^\circ}\)

Compute common ratio:

sin 48° ≈ 0.7431 → 16 / 0.7431 ≈ 21.53

Now:

Side a (opposite 95°):
a = 21.53 × sin 95° ≈ 21.53 × 0.9962 ≈ 21.45

Side c (opposite 37°):
c = 21.53 × sin 37° ≈ 21.53 × 0.6018 ≈ 12.96

✔ Answers:
- Missing angle: 37°
- Side opposite 95°: ≈ 21.45
- Side opposite 37°: ≈ 12.96

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Final Answer:

1. x ≈ 6.21
2. x ≈ 23.04
3. x ≈ 11.60
4. x ≈ 5.84
5. x ≈ 19.59
6. x ≈ 6.20
7. Missing angle: 63°; other sides: ≈8.84 and ≈13.07
8. Missing angle: 37°; other sides: ≈21.45 and ≈12.96