In Exercises 65–68, ABC is a right triangle with the right angle at C. The sides opposite angles A, B, and C are a, b, and c, respectively.


a. Find a and b if c = 2, B = π/3.
b. Find a and c if b = 2, B = π/3.

Verified step by step guidance

1. To solve part (a), we start by using the given information: c = 2 and angle B = π/3. Since ABC is a right triangle, we can use trigonometric ratios. Specifically, we use the sine and cosine functions for angle B.
2. For side a (opposite angle A), we use the sine function: sin(B) = a/c. Plug in the values: sin(π/3) = a/2. Solve for a by multiplying both sides by 2.
3. For side b (adjacent to angle B), we use the cosine function: cos(B) = b/c. Plug in the values: cos(π/3) = b/2. Solve for b by multiplying both sides by 2.
4. To solve part (b), we use the given information: b = 2 and angle B = π/3. We need to find sides a and c. Start with the tangent function: tan(B) = a/b. Plug in the values: tan(π/3) = a/2. Solve for a by multiplying both sides by 2.
5. Finally, use the Pythagorean theorem to find c: a^2 + b^2 = c^2. Substitute the known values of a and b, and solve for c.