Law of Sines and Cosines Notes and Worksheets - Lindsay Bowden - Free Printable

Of course! Let's solve each problem step-by-step using the Law of Sines.

The Law of Sines states that for any triangle with angles A, B, C and opposite sides a, b, c respectively:

> a / sin(A) = b / sin(B) = c / sin(C)

We'll also use the fact that the sum of the interior angles in a triangle is always 180°.

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Problem 1

Given:
- Angles: 53°, 40°
- Side opposite 40°: 5
- Side opposite 53°: x

Step 1: Find the third angle.
Third angle = 180° - 53° - 40° = 87°

Step 2: Apply Law of Sines.
We want to find side `x` opposite the 53° angle.

> x / sin(53°) = 5 / sin(40°)

Step 3: Solve for x.
x = (5 * sin(53°)) / sin(40°)

Using calculator:
- sin(53°) ≈ 0.7986
- sin(40°) ≈ 0.6428

x ≈ (5 * 0.7986) / 0.6428 ≈ 3.993 / 0.6428 ≈ 6.21

✔ Answer: x ≈ 6.21

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Problem 2

Given:
- Angles: 128°, 32°
- Side opposite 32°: 10
- Side opposite 128°: x

Step 1: Find the third angle.
Third angle = 180° - 128° - 32° = 20°

Step 2: Apply Law of Sines.
x / sin(128°) = 10 / sin(32°)

Step 3: Solve for x.
x = (10 * sin(128°)) / sin(32°)

Using calculator:
- sin(128°) = sin(180° - 52°) = sin(52°) ≈ 0.7880
- sin(32°) ≈ 0.5299

x ≈ (10 * 0.7880) / 0.5299 ≈ 7.880 / 0.5299 ≈ 14.87

✔ Answer: x ≈ 14.87

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Problem 3

Given:
- Angles: 99°, 50°
- Side opposite 50°: 9
- Side opposite 99°: x

Step 1: Find the third angle.
Third angle = 180° - 99° - 50° = 31°

Step 2: Apply Law of Sines.
x / sin(99°) = 9 / sin(50°)

Step 3: Solve for x.
x = (9 * sin(99°)) / sin(50°)

Using calculator:
- sin(99°) ≈ 0.9877
- sin(50°) ≈ 0.7660

x ≈ (9 * 0.9877) / 0.7660 ≈ 8.8893 / 0.7660 ≈ 11.61

✔ Answer: x ≈ 11.61

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Problem 4

Given:
- Angles: 62°, 65°
- Side opposite 65°: 6
- Side opposite 62°: x

Step 1: Find the third angle.
Third angle = 180° - 62° - 65° = 53°

Step 2: Apply Law of Sines.
x / sin(62°) = 6 / sin(65°)

Step 3: Solve for x.
x = (6 * sin(62°)) / sin(65°)

Using calculator:
- sin(62°) ≈ 0.8829
- sin(65°) ≈ 0.9063

x ≈ (6 * 0.8829) / 0.9063 ≈ 5.2974 / 0.9063 ≈ 5.84

✔ Answer: x ≈ 5.84

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Problem 5

Given:
- Angles: 110°, 46°
- Side opposite 46°: 15
- Side opposite 110°: x

Step 1: Find the third angle.
Third angle = 180° - 110° - 46° = 24°

Step 2: Apply Law of Sines.
x / sin(110°) = 15 / sin(46°)

Step 3: Solve for x.
x = (15 * sin(110°)) / sin(46°)

Using calculator:
- sin(110°) = sin(70°) ≈ 0.9397
- sin(46°) ≈ 0.7193

x ≈ (15 * 0.9397) / 0.7193 ≈ 14.0955 / 0.7193 ≈ 19.60

✔ Answer: x ≈ 19.60

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Problem 6

Given:
- Angles: 81°, 61°
- Side opposite 81°: 7
- Side opposite 61°: x

Step 1: Find the third angle.
Third angle = 180° - 81° - 61° = 38°

Step 2: Apply Law of Sines.
x / sin(61°) = 7 / sin(81°)

Step 3: Solve for x.
x = (7 * sin(61°)) / sin(81°)

Using calculator:
- sin(61°) ≈ 0.8746
- sin(81°) ≈ 0.9877

x ≈ (7 * 0.8746) / 0.9877 ≈ 6.1222 / 0.9877 ≈ 6.20

✔ Answer: x ≈ 6.20

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Problem 7: Find all missing sides and angles.

Given:
- Angles: 41°, 76°
- Side between them (opposite the unknown angle): 12

Step 1: Find the third angle.
Third angle = 180° - 41° - 76° = 63°

Step 2: Label sides.
Let’s say:
- Side `a` = 12 (opposite 63°)
- Side `b` opposite 41° → let’s call it `y`
- Side `c` opposite 76° → let’s call it `z`

Step 3: Apply Law of Sines.
12 / sin(63°) = y / sin(41°) = z / sin(76°)

First, compute 12 / sin(63°):
- sin(63°) ≈ 0.8910
→ 12 / 0.8910 ≈ 13.468

Now find `y` (opposite 41°):
y = 13.468 * sin(41°) ≈ 13.468 * 0.6561 ≈ 8.84

Now find `z` (opposite 76°):
z = 13.468 * sin(76°) ≈ 13.468 * 0.9703 ≈ 13.07

✔ Answers:
- Third 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°, 48°
- Side between them (opposite the unknown angle): 16

Step 1: Find the third angle.
Third angle = 180° - 95° - 48° = 37°

Step 2: Label sides.
Let’s say:
- Side `a` = 16 (opposite 37°)
- Side `b` opposite 95° → let’s call it `y`
- Side `c` opposite 48° → let’s call it `z`

Step 3: Apply Law of Sines.
16 / sin(37°) = y / sin(95°) = z / sin(48°)

First, compute 16 / sin(37°):
- sin(37°) ≈ 0.6018
→ 16 / 0.6018 ≈ 26.587

Now find `y` (opposite 95°):
y = 26.587 * sin(95°) ≈ 26.587 * 0.9962 ≈ 26.49

Now find `z` (opposite 48°):
z = 26.587 * sin(48°) ≈ 26.587 * 0.7431 ≈ 19.76

✔ Answers:
- Third angle: 37°
- Side opposite 95°: ≈ 26.49
- Side opposite 48°: ≈ 19.76

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## ✔ Final Answers Summary:

1. x ≈ 6.21
2. x ≈ 14.87
3. x ≈ 11.61
4. x ≈ 5.84
5. x ≈ 19.60
6. x ≈ 6.20
7. Missing angle: 63°; Missing sides: ≈8.84 and ≈13.07
8. Missing angle: 37°; Missing sides: ≈26.49 and ≈19.76

Let me know if you’d like these rounded to different decimal places or shown with work on paper format!