Law Of Sines Worksheet - Fill Online, Printable, Fillable, Blank ... - Free Printable

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

The Law of Sines says:

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

We’ll use this to find missing sides or angles in each triangle.

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Problem 1: Triangle with angles 30°, 25°, side 16 opposite 30°

First, find the third angle:
Sum of angles in a triangle = 180°
So, third angle = 180° - 30° - 25° = 125°

Now, label:
- Angle A = 30° → side a = 16 (opposite)
- Angle B = 25° → side b = ? (we’ll call it x)
- Angle C = 125° → side c = ? (we’ll call it y)

Use Law of Sines:

\(\frac{\sin 30^\circ}{16} = \frac{\sin 25^\circ}{x}\)

Solve for x:

\(x = \frac{16 \cdot \sin 25^\circ}{\sin 30^\circ}\)

Calculate:
- sin 25° ≈ 0.4226
- sin 30° = 0.5

→ \(x = \frac{16 \cdot 0.4226}{0.5} = \frac{6.7616}{0.5} = 13.5232\)

Round to nearest tenth: x ≈ 13.5

Now find y (side opposite 125°):

\(\frac{\sin 30^\circ}{16} = \frac{\sin 125^\circ}{y}\)

→ \(y = \frac{16 \cdot \sin 125^\circ}{\sin 30^\circ}\)

sin 125° = sin(180° - 55°) = sin 55° ≈ 0.8192

→ \(y = \frac{16 \cdot 0.8192}{0.5} = \frac{13.1072}{0.5} = 26.2144\)

Round to nearest tenth: y ≈ 26.2

✔ Final answers for #1:
Missing side opposite 25°: 13.5
Missing side opposite 125°: 26.2

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Problem 2: Triangle with angles 25°, 40°, side 40 opposite 40°

Third angle = 180° - 25° - 40° = 115°

Label:
- Angle A = 25° → side a = ?
- Angle B = 40° → side b = 40
- Angle C = 115° → side c = ?

Use Law of Sines:

\(\frac{\sin 40^\circ}{40} = \frac{\sin 25^\circ}{a}\)

→ \(a = \frac{40 \cdot \sin 25^\circ}{\sin 40^\circ}\)

sin 25° ≈ 0.4226
sin 40° ≈ 0.6428

→ \(a = \frac{40 \cdot 0.4226}{0.6428} = \frac{16.904}{0.6428} ≈ 26.3\)

Now find side c (opposite 115°):

\(\frac{\sin 40^\circ}{40} = \frac{\sin 115^\circ}{c}\)

sin 115° = sin(180° - 65°) = sin 65° ≈ 0.9063

→ \(c = \frac{40 \cdot 0.9063}{0.6428} = \frac{36.252}{0.6428} ≈ 56.4\)

✔ Final answers for #2:
Side opposite 25°: 26.3
Side opposite 115°: 56.4

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Problem 3: Triangle with angles 75°, 40°, side 40 opposite 40°

Wait — actually, looking at the diagram:
Angle at top is 75°, bottom right is 40°, so bottom left must be:

180° - 75° - 40° = 65°

Side labeled “40” is opposite the 40° angle? Wait — no! Let’s check carefully.

In the drawing:
- Top angle = 75°
- Bottom right angle = 40°
- Side between them (bottom side) = 40 → that side is opposite the TOP angle (75°)!
Actually, let’s label properly.

Assume:
- Angle A = 75° → side a = ? (opposite)
- Angle B = 40° → side b = ? (opposite)
- Angle C = 65° → side c = 40 (given, opposite 65°)

Wait — if the side labeled “40” is the base, and the two base angles are 75° and 40°, then the side opposite the 75° angle is NOT given — actually, the side opposite the 65° angle is the one we’re calling “40”.

Better to assign:

Let’s say:
- Angle at vertex A = 75° → side a = BC = ?
- Angle at vertex B = 40° → side b = AC = ?
- Angle at vertex C = 65° → side c = AB = 40

Then Law of Sines:

\(\frac{\sin 65^\circ}{40} = \frac{\sin 75^\circ}{a} = \frac{\sin 40^\circ}{b}\)

Find side a (opposite 75°):

\(a = \frac{40 \cdot \sin 75^\circ}{\sin 65^\circ}\)

sin 75° ≈ 0.9659
sin 65° ≈ 0.9063

→ \(a = \frac{40 \cdot 0.9659}{0.9063} = \frac{38.636}{0.9063} ≈ 42.6\)

Find side b (opposite 40°):

\(b = \frac{40 \cdot \sin 40^\circ}{\sin 65^\circ} = \frac{40 \cdot 0.6428}{0.9063} = \frac{25.712}{0.9063} ≈ 28.4\)

✔ Final answers for #3:
Side opposite 75°: 42.6
Side opposite 40°: 28.4

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Problem 4: Triangle with angles 140°, and two other angles unknown, but side 101 opposite 140°, and another side 50 adjacent?

Wait — look at diagram:
It shows an obtuse triangle with angle 140°, side opposite it is 101, and another side is 50 (adjacent to 140°). We need to find the other angles and side.

Label:
- Angle A = 140° → side a = 101 (opposite)
- Side b = 50 → opposite angle B = ?
- Side c = ? → opposite angle C = ?

First, use Law of Sines to find angle B:

\(\frac{\sin 140^\circ}{101} = \frac{\sin B}{50}\)

→ \(\sin B = \frac{50 \cdot \sin 140^\circ}{101}\)

sin 140° = sin(180° - 40°) = sin 40° ≈ 0.6428

→ \(\sin B = \frac{50 \cdot 0.6428}{101} = \frac{32.14}{101} ≈ 0.3182\)

Now find angle B:
B = arcsin(0.3182) ≈ 18.5°

Then angle C = 180° - 140° - 18.5° = 21.5°

Now find side c (opposite angle C):

\(\frac{\sin 140^\circ}{101} = \frac{\sin 21.5^\circ}{c}\)

→ \(c = \frac{101 \cdot \sin 21.5^\circ}{\sin 140^\circ}\)

sin 21.5° ≈ 0.3665
sin 140° ≈ 0.6428

→ \(c = \frac{101 \cdot 0.3665}{0.6428} = \frac{37.0165}{0.6428} ≈ 57.6\)

✔ Final answers for #4:
Angle opposite side 50: 18.5°
Other angle: 21.5°
Missing side: 57.6

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Problem 5: Find perimeter of △DEF

Given:
- Angle D = ? (not given directly)
- Angle E = 38°
- Angle F = 30°
- Side DE = 16 → this is side opposite angle F (since F is at vertex F, opposite side is DE)
Wait — let’s label properly.

Triangle DEF:
- Vertices: D, E, F
- Given: angle at E = 38°, angle at F = 30°
→ So angle at D = 180° - 38° - 30° = 112°

Sides:
- Side opposite D = EF = ?
- Side opposite E = DF = 16 (given as "16" on side from D to F? Wait — diagram says:

Looking at diagram:
Point D connected to E (length 16), E to F (unknown), F to D (unknown).
Angles: at E = 38°, at F = 30° → so side opposite E is DF, side opposite F is DE.

Actually, standard labeling:
- Side opposite angle D is EF
- Side opposite angle E is DF
- Side opposite angle F is DE

Given: DE = 16 → that’s side opposite angle F (which is 30°)

So:
- Angle F = 30° → side f = DE = 16
- Angle E = 38° → side e = DF = ?
- Angle D = 112° → side d = EF = ?

Use Law of Sines:

\(\frac{\sin 30^\circ}{16} = \frac{\sin 38^\circ}{e} = \frac{\sin 112^\circ}{d}\)

First, find e (DF):

\(e = \frac{16 \cdot \sin 38^\circ}{\sin 30^\circ}\)

sin 38° ≈ 0.6157
sin 30° = 0.5

→ \(e = \frac{16 \cdot 0.6157}{0.5} = \frac{9.8512}{0.5} = 19.7024 ≈ 19.7\)

Now find d (EF):

\(d = \frac{16 \cdot \sin 112^\circ}{\sin 30^\circ}\)

sin 112° = sin(180° - 68°) = sin 68° ≈ 0.9272

→ \(d = \frac{16 \cdot 0.9272}{0.5} = \frac{14.8352}{0.5} = 29.6704 ≈ 29.7\)

Perimeter = DE + EF + FD = 16 + 29.7 + 19.7 = 65.4

✔ Final answer for #5: Perimeter = 65.4

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## Final Answer:

1) Missing sides: 13.5, 26.2
2) Missing sides: 26.3, 56.4
3) Missing sides: 42.6, 28.4
4) Missing angles: 18.5°, 21.5°; missing side: 57.6
5) Perimeter of △DEF: 65.4