In triangle ABC, if angle A = 30 degrees and angle B = 60 degrees, what is the l

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Question: In triangle ABC, if angle A = 30 degrees and angle B = 60 degrees, what is the length of side a opposite angle A if side b = 10?

Options:

Correct Answer: 5

Solution:

Using the sine rule: a/sin(A) = b/sin(B) => a = b * sin(A)/sin(B) = 10 * sin(30)/sin(60) = 10 * 0.5/(√3/2) = 5.

In triangle ABC, if angle A = 30 degrees and angle B = 60 degrees, what is the length of side a opposite angle A if side b = 10?

In triangle ABC, if angle A = 30 degrees and angle B = 60 degrees, what is the length of side a opposite angle A if side b = 10?

Step 1: Identify the angles in triangle ABC. We have angle A = 30 degrees and angle B = 60 degrees.
Step 2: Use the sine rule, which states that the ratio of the length of a side to the sine of its opposite angle is the same for all three sides of the triangle.
Step 3: Write the sine rule for sides a and b: a/sin(A) = b/sin(B).
Step 4: Substitute the known values into the sine rule: a/sin(30) = 10/sin(60).
Step 5: Rearrange the equation to solve for a: a = b * sin(A) / sin(B).
Step 6: Substitute b = 10, sin(30) = 0.5, and sin(60) = √3/2 into the equation: a = 10 * sin(30) / sin(60).
Step 7: Calculate the value: a = 10 * 0.5 / (√3/2).
Step 8: Simplify the calculation: a = 10 * 0.5 * (2/√3) = 10 / √3.
Step 9: Calculate the final value of a, which is approximately 5.77.

Sine Rule – The sine rule relates the lengths of the sides of a triangle to the sines of its angles, allowing for the calculation of unknown side lengths when certain angles and side lengths are known.
Triangle Angle Sum – The sum of the angles in a triangle is always 180 degrees, which can help verify the angles given in the problem.