Law of Sines Worksheet

The law of sines is a fundamental concept in trigonometry, used to find the lengths of the sides of a triangle when we know the measure of two angles and one side, or when we know the measure of two sides and one of the angles is not the included angle. It states that for any given triangle with sides of lengths a, b, and c, and opposite angles A, B, and C, respectively, the following equation holds:

a / sin(A) = b / sin(B) = c / sin©

This law can be applied to various types of triangles, including right triangles, oblique triangles, and even triangles where all sides and angles are unknown, provided enough information is given.

To apply the law of sines, follow these steps:

1. Identify what you’re trying to find (either an angle or a side).
2. Use the given information (either two angles and one side, or two sides and a non-included angle) to set up the equation according to the law of sines formula.
3. Solve the equation for the unknown.

Here are a few examples of how to apply the law of sines in different scenarios:

Example 1: Finding a Side

Given a triangle with angle A = 30 degrees, angle B = 60 degrees, and side a = 5 inches, find side b.

First, find angle C since the sum of angles in a triangle is 180 degrees: C = 180 - (A + B) = 180 - (30 + 60) = 90 degrees.

Then apply the law of sines: a / sin(A) = b / sin(B)

Substitute the known values: 5 / sin(30) = b / sin(60)

To find b: b = 5 * sin(60) / sin(30)

Calculating the value of b: b = 5 * 0.866 / 0.5 b = 8.66 inches

Example 2: Finding an Angle

Given a triangle with side a = 10 cm, side b = 15 cm, and angle A = 45 degrees, find angle B.

Using the law of sines: a / sin(A) = b / sin(B)

Substitute the known values: 10 / sin(45) = 15 / sin(B)

To find sin(B): sin(B) = 15 * sin(45) / 10 sin(B) = 15 * 0.707 / 10 sin(B) = 1.0605

Since sin(B) cannot be greater than 1, there seems to be an error in the calculation or the given values are not suitable for a real triangle.

Let’s correct the approach by recognizing that the law of sines requires that the ratio of the sides to the sines of their opposite angles must be equal for all three sides and angles. The mistake in calculation indicates a misunderstanding of the given values or an error in the math.

Correcting the mistake, we should recognize that if we’re given two sides and an angle not between them, we should use the law of sines to find another angle or side, ensuring our calculations align with the principles of trigonometry and the specifics of the triangle in question.

Example 3: Real-World Application

In navigating, the law of sines can be used to determine distances or angles between landmarks when other measurements are known. For instance, if you’re sailing and know the distance to a lighthouse (side a), the angle from your position to the lighthouse (angle A), and another angle in the triangle formed by your position, the lighthouse, and another landmark (angle B), you can use the law of sines to calculate the distance to the other landmark or the remaining angle.

Given: - Distance to the lighthouse (a) = 5 miles - Angle A (from your position to the lighthouse) = 30 degrees - Angle B (at the lighthouse towards the other landmark) = 60 degrees

To find the distance to the other landmark (side b): First, calculate angle C: C = 180 - (A + B) = 180 - (30 + 60) = 90 degrees

Then apply the law of sines: 5 / sin(30) = b / sin(60)

Solving for b: b = 5 * sin(60) / sin(30) b = 5 * 0.866 / 0.5 b = 8.66 miles

Practice Questions

1. In a triangle, angle A = 40 degrees, angle B = 80 degrees, and side a = 7 cm. Find side b.

2. Given a triangle with side a = 9 inches, side b = 12 inches, and angle A = 50 degrees, find angle B.

3. A pilot is flying towards a tower. The angle from the pilot’s line of sight to the top of the tower is 20 degrees. If the pilot is 1000 feet away from the point directly below the top of the tower and the angle at the top of the tower to the pilot’s current position is 70 degrees, how far is the pilot from the tower?

Solutions

1. Find angle C first: C = 180 - (A + B) = 180 - (40 + 80) = 60 degrees

Then apply the law of sines: 7 / sin(40) = b / sin(80)

b = 7 * sin(80) / sin(40)

Calculating b: b = 7 * 0.9848 / 0.6428 b = 10.78 cm

1. Apply the law of sines directly: 9 / sin(50) = 12 / sin(B)

To find sin(B): sin(B) = 12 * sin(50) / 9 sin(B) = 12 * 0.766 / 9 sin(B) = 1.025 (This calculation indicates an error since sin(B) must be ≤1. The initial values given may not form a valid triangle under the conditions described, or there’s a calculation error.)

1. First, find the angle at the pilot’s position towards the tower (angle C): C = 180 - (20 + 70) = 90 degrees

This forms a right triangle. The distance from the pilot to the point directly below the top of the tower (side a) = 1000 feet, and angle A = 20 degrees.

Using trigonometry for right triangles (since we have a right angle): tan(A) = opposite / adjacent tan(20) = height of the tower / 1000

To find the height (opposite to angle A): height = 1000 * tan(20) height = 1000 * 0.364 height = 364 feet

To find the distance from the pilot to the tower (hypotenuse), use the Pythagorean theorem or recognize that in a right triangle, the law of sines simplifies to the basic trigonometric ratios.