Law Of Cosines Practice Problems

Mastering the Law of Cosines: Practice Problems and Solutions

The Law of Cosines is a fundamental theorem in trigonometry, extending the Pythagorean theorem to non-right-angled triangles. It provides a powerful tool for solving problems involving the sides and angles of any triangle, making it indispensable in fields like surveying, navigation, and engineering. This article delves into the Law of Cosines, providing a range of practice problems with detailed solutions, catering to various skill levels. Understanding this law is crucial for anyone pursuing a deeper understanding of trigonometry and its applications. We'll cover various scenarios, from finding missing sides to calculating unknown angles, solidifying your grasp of this essential mathematical concept.

Understanding the Law of Cosines

The Law of Cosines states the relationship between the lengths of the sides of a triangle and the cosine of one of its angles. For any triangle with sides a, b, and c, and angles A, B, and C opposite to their respective sides, the law is expressed in three equivalent forms:

• a² = b² + c² - 2bc cos A
• b² = a² + c² - 2ac cos B
• c² = a² + b² - 2ab cos C

Essentially, this law allows us to find:

• A missing side length: If we know two side lengths and the included angle, we can calculate the length of the third side.
• A missing angle: If we know all three side lengths, we can calculate any of the angles.

Remember that the angle used in the formula is always the angle opposite the side being solved for. This subtle detail is crucial for correctly applying the Law of Cosines.

Practice Problems: Finding Missing Sides

Let's start with problems focusing on calculating the length of a missing side. Each problem will be followed by a step-by-step solution.

Problem 1:

A triangle has sides b = 7 cm and c = 10 cm, and the angle between them, A, is 60°. Find the length of side a.

Solution 1:

We use the formula: a² = b² + c² - 2bc cos A

Substitute the given values:

a² = 7² + 10² - 2(7)(10) cos 60°

a² = 49 + 100 - 140 (0.5)

a² = 149 - 70

a² = 79

a = √79 ≈ 8.89 cm

Problem 2:

In a triangle, sides a = 5 inches and c = 12 inches, and angle B = 135°. Find the length of side b.

Solution 2:

We use the formula: b² = a² + c² - 2ac cos B

Substitute the given values:

b² = 5² + 12² - 2(5)(12) cos 135°

b² = 25 + 144 - 120 (-√2/2)

b² = 169 + 60√2

b² ≈ 169 + 84.85

b² ≈ 253.85

b ≈ √253.85 ≈ 15.93 inches

Problem 3:

A triangular field has sides of length 25 meters and 30 meters. The angle between these sides is 120°. Calculate the length of the third side.

Solution 3:

Let the known sides be a = 25 m and b = 30 m, and the angle between them be C = 120°. We need to find c.

c² = a² + b² - 2ab cos C

c² = 25² + 30² - 2(25)(30) cos 120°

c² = 625 + 900 - 1500 (-0.5)

c² = 1525 + 750

c² = 2275

c = √2275 ≈ 47.7 meters

Practice Problems: Finding Missing Angles

Now let's tackle problems where we need to determine the value of an unknown angle.

Problem 4:

A triangle has sides a = 6, b = 8, and c = 10. Find angle A.

Solution 4:

We use the formula: a² = b² + c² - 2bc cos A

Rearrange the formula to solve for cos A:

cos A = (b² + c² - a²) / 2bc

Substitute the given values:

cos A = (8² + 10² - 6²) / (2 * 8 * 10)

cos A = (64 + 100 - 36) / 160

cos A = 128 / 160 = 0.8

A = cos⁻¹(0.8) ≈ 36.87°

Problem 5:

A triangle has sides a = 9, b = 12, and c = 15. Find angle B.

Solution 5:

We use the formula: b² = a² + c² - 2ac cos B

Rearrange to solve for cos B:

cos B = (a² + c² - b²) / 2ac

Substitute the given values:

cos B = (9² + 15² - 12²) / (2 * 9 * 15)

cos B = (81 + 225 - 144) / 270

cos B = 162 / 270 = 0.6

B = cos⁻¹(0.6) ≈ 53.13°

Problem 6:

The sides of a triangle are 13 cm, 14 cm, and 15 cm. Find the largest angle in the triangle.

Solution 6:

The largest angle will be opposite the longest side. Therefore, we'll find angle C (opposite side c = 15 cm).

c² = a² + b² - 2ab cos C

cos C = (a² + b² - c²) / 2ab

cos C = (13² + 14² - 15²) / (2 * 13 * 14)

cos C = (169 + 196 - 225) / 364

cos C = 140 / 364 ≈ 0.3846

C = cos⁻¹(0.3846) ≈ 67.38°

Advanced Practice Problems

These problems incorporate more complex scenarios and require a deeper understanding of the Law of Cosines and potentially other trigonometric concepts.

Problem 7:

Two ships leave a port at the same time. Ship A travels at 15 km/h on a bearing of 060°. Ship B travels at 20 km/h on a bearing of 110°. After 2 hours, what is the distance between the two ships?

Solution 7:

First, find the distances traveled by each ship:

Distance A = 15 km/h * 2 h = 30 km Distance B = 20 km/h * 2 h = 40 km

The angle between their paths is 110° - 60° = 50°. We can now use the Law of Cosines to find the distance between them (let's call it 'c'):

c² = a² + b² - 2ab cos C

c² = 30² + 40² - 2(30)(40) cos 50°

c² = 900 + 1600 - 2400 cos 50°

c² ≈ 2500 - 1536.

c² ≈ 964

c ≈ √964 ≈ 31.05 km

Problem 8:

A surveyor measures the distance between two points, A and B, to be 500 meters. From point A, the surveyor measures the angle to a third point, C, to be 75°. The distance from A to C is 600 meters. Find the distance from B to C.

Solution 8:

We have a triangle ABC with sides a (BC), b (AC = 600m), and c (AB = 500m). Angle A is 75°.

We can use the Law of Cosines to find a:

a² = b² + c² - 2bc cos A

a² = 600² + 500² - 2(600)(500) cos 75°

a² ≈ 360000 + 250000 - 600000(0.2588)

a² ≈ 610000 - 155280

a² ≈ 454720

a ≈ √454720 ≈ 674.3 meters

Frequently Asked Questions (FAQ)

Q: When should I use the Law of Cosines instead of the Law of Sines?

A: The Law of Sines is best suited when you know two angles and one side, or two sides and an angle opposite one of them. The Law of Cosines is preferable when you know three sides or two sides and the included angle.

Q: Can I use the Law of Cosines to solve right-angled triangles?

A: Yes, but it's generally more efficient to use the Pythagorean theorem and basic trigonometric functions (sin, cos, tan) for right-angled triangles. The Law of Cosines will still work, but it adds unnecessary complexity.

Q: What if I get a negative value when calculating the cosine of an angle?

A: A negative cosine value indicates that the angle is obtuse (greater than 90°). Your calculation is likely correct. Use the inverse cosine function (cos⁻¹) to find the angle's measure.

Q: What are some common mistakes to avoid when using the Law of Cosines?

A: Common errors include using the wrong formula (confusing a, b, and c), incorrectly substituting values, and forgetting to take the square root when solving for a side length. Always double-check your work and ensure you're using the correct formula and substituting the correct values.

Conclusion

The Law of Cosines is a powerful tool for solving various trigonometric problems. By practicing a diverse range of problems, from simple applications to more complex scenarios, you will build a solid understanding of this fundamental principle. Remember to carefully identify the given information, choose the appropriate formula, substitute values accurately, and carefully interpret your results. Through consistent practice, you will master the Law of Cosines and unlock its applications in various mathematical and real-world contexts. Continue to practice and challenge yourself with different problem types to solidify your understanding and build confidence in your ability to solve even the most intricate trigonometric puzzles. Remember that understanding the underlying principles and practicing consistently is key to mastering the Law of Cosines.