The Law of Sines is a fundamental principle in trigonometry that relates the sides of a triangle to the sines of its angles. It is especially useful for solving triangles, whether they are right-angled or not. The Law of Cosines is another important principle in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is particularly useful for solving triangles when you know two sides and the included angle or all three sides.

Law of Sines

The Law of Sines relates the sides of a triangle to its angles. It states that for any triangle:

[Tex]\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c} [/Tex]​

Where:

• a, b, and care the lengths of the sides opposite the angles ????A, ????B, and ????C, respectively.
• sin represents the sine function, which relates to the angle in a right-angled triangle.

Read More: Law of Sines – Formula, Definition, Examples, Proof

Law of Cosines

The Law of Cosines is used to find a side or angle in any triangle. It states that:

[Tex]c^2 = a^2 + b^2 – 2ab \cos(C) \\ b^2 = a^2 + c^2 – 2ac \cos(B) \\ a^2 = b^2 + c^2 – 2bc \cos(A) [/Tex]

Where:

• a, b, and c are the lengths of the sides of the triangle.
• C is the angle opposite the side c.
• cos⁡ represents the cosine function, which relates to the angle in a right-angled triangle.

Read More: Law of Cosine – Formula, Proof & Difference with Sin Rule

Formula – Law of Sines and Cosines

Law of Sines

[Tex]\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}[/Tex]

Where:

• a, b and c are the lengths of the sides of the triangle.
• A, B and C are the angles opposite those sides, respectively.

Law of Cosines

The Law of Cosines is used to find the length of a side or the measure of an angle in any triangle. The formulas are:

To find the length of a side:

[Tex]c^2 = a^2 + b^2 – 2ab \cos(C) \\ b^2 = a^2 + c^2 – 2ac \cos(B) \\ a^2 = b^2 + c^2 – 2bc \cos(A)[/Tex]

Where:

• a, b, and c are the lengths of the sides of the triangle.
• A, B, and C are the angles opposite those sides, respectively.

To find the measure of an angle:

• cos A = [b² + c² – a²]/2bc
• cos B = [a² + c² – b²]/2ac
• cos C = [b² + a² – c²]/2ab

These formulas help in solving triangles when given some combination of sides and angles.

Practical Examples – Law of Sines and Cosines

Example 1: Navigation (Law of Sines)

A ship at sea sights a lighthouse at an angle of elevation of 30°. If the lighthouse is known to be 2 kilometers offshore, the ship can use the Law of Sines to calculate its distance from the lighthouse, assuming the ship’s position is known relative to a fixed point.

Example 2: Triangulation (Law of Sines)

In surveying, if two surveyors are positioned at known distances from each other and both measure the angle to a third point, the Law of Sines can be used to determine the distance to the third point from each surveyor, assuming the angles are measured accurately.

Example 3: Structure Design (Law of Cosines)

When designing a bridge span, engineers may need to calculate the length of a diagonal support beam between two known points on either side of the span. The Law of Cosines can be applied to determine this length, considering the angle and distances involved.

Example 4: Celestial Navigation (Law of Sines)

In celestial navigation, sailors historically used the Law of Sines to determine their latitude based on the angle of the sun or stars above the horizon at noon or at specific times of the day.

Read: Law of Sine and Cosine: Formula and Examples

Practice Problems on Law of Sines and Cosines: Solved

Problem: In triangle ABC, a = 6, b = 8, and C = 30°. Find angle A.

sin A / 6 = sin 30° / 8

sin A = (6 sin 30°) / 8

A = sin^-1((6 * 0.5) / 8) ≈ 22.0°

Problem: In triangle PQR, p = 10, q = 12, and R = 100°. Find r.

r / sin 100° = 10 / sin P = 12 / sin Q

r = (10 sin 100°) / sin P

r ≈ 11.7 (using a calculator)

Problem: In triangle XYZ, x = 5, y = 7, and z = 8. Find angle X.

cos X = (y² + z² – x²) / (2yz)

cos X = (7² + 8² – 5²) / (2 x 7 x 8)

cos X = (49 + 64 – 25) / 112 = 88 / 112

X = cos^-1 (88/112) ≈ 38.2°

Problem: In triangle DEF, d = 9, e = 11, and F = 50°. Find f.

f² = d² + e² – 2de cos F

f² = 9² + 11² – 2(9)(11)cos 50°

f² = 81 + 121 – 198 x 0.6428 ≈ 74.73

f ≈ 8.64

Problem: In triangle ABC, A = 60°, B = 45°, and c = 10. Find a.

a / sin 60° = 10 / sin 75°

a = (10 sin 60°) / sin 75°

a ≈ 9.13

Problem: In triangle LMN, L = 30°, M = 60°, and l = 12. Find n.

12 / sin 60° = n / sin 30°

n = (12 sin 30°) / sin 60°

n = (12 x 0.5) / (√3/2) = 12 / √3

n ≈ 6.93

Problem: In triangle PQR, p = 8, q = 10, and r = 12. Find angle R.

cos R = (p² + q² – r²) / (2pq)

cos R = (8² + 10² – 12²) / (2 x 8 x 10)

cos R = (64 + 100 – 144) / 160 = 20 / 160 = 1/8

R = cos^-1 (1/8) ≈ 82.8°

Problem: In triangle ABC, a = 5, b = 7, and C = 40°. Find c.

c² = a² + b² – 2ab cos C

c² = 5² + 7² – 2(5)(7)cos 40°

c² = 25 + 49 – 70 x 0.7660 ≈ 20.38

c ≈ 4.51

Problem: In triangle XYZ, X = 50°, Y = 70°, and z = 15. Find x.

x / sin 50° = 15 / sin 60°

x = (15 sin 50°) / sin 60°

x ≈ 13.36

Problem: In triangle DEF, d = 6, e = 8, and f = 9. Find angle D.

cos D = (e² + f² – d²) / (2ef)

cos D = (8² + 9² – 6²) / (2 x 8 x 9)

cos D = (64 + 81 – 36) / 144 = 109 / 144

D = cos^-1 (109/144) ≈ 41.4°

Practice Problems on Law of Sines and Cosines: Unsolved

Law of Sines Problems:

1. In triangle ABC, angle A = 45°, angle B = 60°, and side a = 10 cm. Find the length of side b.
2. In triangle PQR, side p = 8 cm, side q = 12 cm, and angle P = 30°. Find angle Q.
3. In triangle XYZ, angle X = 55°, angle Y = 75°, and side z = 15 cm. Find the lengths of sides x and y.

Law of Cosines Problems:

1. In triangle DEF, side d = 6 cm, side e = 8 cm, and angle F = 120°. Find the length of side f.
2. In triangle GHI, side g = 10 cm, side h = 12 cm, and side i = 14 cm. Find the measure of angle G.
3. In triangle JKL, side j = 15 cm, angle K = 45°, and angle L = 60°. Find the length of side k.

Mixed Problems (using both laws):

1. In triangle MNO, side m = 20 cm, side n = 25 cm, and angle O = 100°. Find the measures of angles M and N.
2. In triangle STU, side s = 18 cm, side t = 24 cm, and side u = 30 cm. Find all three angles of the triangle.
3. In triangle VWX, angle V = 40°, side w = 12 cm, and side x = 15 cm. Find the length of side v and the measure of angle W.
4. In triangle ABC, side a = 8 cm, side b = 10 cm, and angle C = 50°. Find the length of side c and the measures of angles A and B.

FAQs – Law of Sines and Cosines

When should I use the Law of Sines vs. the Law of Cosines?

Use the Law of Sines when you know two angles and any side, or two sides and a non-included angle. Use the Law of Cosines when you know two sides and the included angle, or all three sides.

What information do I need to solve a triangle using these laws?

You need at least three pieces of information about the triangle, including at least one side length.

Can I use the Law of Sines to find an angle if I know all three sides?

No, in this case you should use the Law of Cosines. The Law of Sines requires at least one known angle.

What’s the most common mistake in these problems?

Forgetting to consider the ambiguous case when using the Law of Sines, which can sometimes result in two possible solutions.

How do I know if my answer is reasonable?

Check if your angles sum to 180°, and use the triangle inequality theorem (the sum of any two sides must be greater than the third side).