Solving Triangles with Ease: Law of Sines Worksheet Answers

Understanding the Law of Sines

The Law of Sines is a fundamental concept in trigonometry that helps us solve triangles with ease. It states that for any given triangle with angles A, B, and C, and opposite sides a, b, and c, the ratio of the length of a side to the sine of its opposite angle is constant. Mathematically, this can be represented as:

a / sin(A) = b / sin(B) = c / sin©

This law is useful in solving triangles when we know two angles and one side, or two sides and one angle. In this article, we will explore the Law of Sines in more detail and provide a worksheet with answers to help you practice and reinforce your understanding.

How to Use the Law of Sines

To use the Law of Sines, follow these steps:

1. Identify the given information: Write down the given angles and sides of the triangle.
2. Identify the unknown information: Determine what you need to find, whether it’s an angle or a side.
3. Choose the correct equation: Select the equation that involves the given and unknown information.
4. Plug in the values: Substitute the given values into the equation.
5. Solve for the unknown: Use algebraic manipulation to solve for the unknown value.

Worksheet with Answers

Here’s a worksheet with 5 problems to help you practice using the Law of Sines:

Problem 1

In triangle ABC, angle A = 30°, angle B = 60°, and side a = 10 cm. Find side b.

Solution

Using the Law of Sines, we can write:

10 / sin(30°) = b / sin(60°)

Plugging in the values, we get:

10 / 0.5 = b / 0.866

Solving for b, we get:

b = 17.32 cm

Problem 2

In triangle DEF, angle D = 45°, angle E = 90°, and side d = 8 cm. Find side e.

Solution

Using the Law of Sines, we can write:

8 / sin(45°) = e / sin(90°)

Plugging in the values, we get:

8 / 0.707 = e / 1

Solving for e, we get:

e = 11.31 cm

Problem 3

In triangle GHI, angle G = 60°, angle H = 80°, and side g = 12 cm. Find angle I.

Solution

Using the Law of Sines, we can write:

12 / sin(60°) = h / sin(80°)

Plugging in the values, we get:

12 / 0.866 = h / 0.984

Solving for h, we get:

h = 13.28 cm

Using the fact that the sum of angles in a triangle is 180°, we can find angle I:

I = 180° - 60° - 80° = 40°

Problem 4

In triangle JKL, angle J = 30°, angle K = 90°, and side j = 6 cm. Find side k.

Solution

Using the Law of Sines, we can write:

6 / sin(30°) = k / sin(90°)

Plugging in the values, we get:

6 / 0.5 = k / 1

Solving for k, we get:

k = 12 cm

Problem 5

In triangle MNO, angle M = 45°, angle N = 60°, and side m = 9 cm. Find side n.

Solution

Using the Law of Sines, we can write:

9 / sin(45°) = n / sin(60°)

Plugging in the values, we get:

9 / 0.707 = n / 0.866

Solving for n, we get:

n = 11.43 cm

📝 Note: These problems are meant to be solved using the Law of Sines. Make sure to show your work and use the correct equations to arrive at the answers.

In conclusion, the Law of Sines is a powerful tool for solving triangles. By understanding the law and practicing with worksheets, you can become more confident and proficient in your ability to solve triangles with ease. Remember to show your work, use the correct equations, and double-check your answers to ensure accuracy.