In triangle ABC, if angle A = 45° and side a = 10, what is the length of side b

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Question: In triangle ABC, if angle A = 45° and side a = 10, what is the length of side b if angle B = 60°?

Options:

Correct Answer: 8.66

Solution:

Using the Law of Sines: b/a = sin(B)/sin(A) => b = a * (sin(B)/sin(A)) = 10 * (√3/2)/(√2/2) = 10 * √3/√2 = 10 * 8.66/10 = 8.66.

In triangle ABC, if angle A = 45° and side a = 10, what is the length of side b if angle B = 60°?

In triangle ABC, if angle A = 45° and side a = 10, what is the length of side b if angle B = 60°?

Step 1: Identify the given information. We have a triangle ABC where angle A = 45°, side a = 10, and angle B = 60°.
Step 2: Use the Law of Sines formula, which states that the ratio of a side to the sine of its opposite angle is the same for all sides. The formula is: b/a = sin(B)/sin(A).
Step 3: Substitute the known values into the formula. We have b/10 = sin(60°)/sin(45°).
Step 4: Calculate sin(60°) and sin(45°). We know that sin(60°) = √3/2 and sin(45°) = √2/2.
Step 5: Substitute these values into the equation: b/10 = (√3/2) / (√2/2).
Step 6: Simplify the right side of the equation. This gives us b/10 = (√3/2) * (2/√2) = √3/√2.
Step 7: Multiply both sides by 10 to solve for b: b = 10 * (√3/√2).
Step 8: Simplify the expression. This can be calculated as b = 10 * (√3/√2) = 10 * (√3 * √2) / 2 = 10 * (√6 / 2) = 5√6.
Step 9: Calculate the numerical value of b. Using a calculator, 5√6 is approximately 12.25.

Law of Sines – The Law of Sines relates the lengths of sides of a triangle to the sines of its angles, allowing for the calculation of unknown side lengths when certain angles and sides are known.
Angle Relationships – Understanding the relationship between angles in a triangle, particularly how the sum of angles in a triangle equals 180°.