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Solve the following triangle.

B = 60°, C = 40°, b = 7

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Answer:

To solve the triangle, we can start by using the Law of Sines to find the other two angles:

a/sin A = b/sin B = c/sin C

We know B = 60°, C = 40°, and b = 7, so we can solve for a and c:

a/sin A = 7/sin 60°

a = (7 sin A)/sin 60°

c/sin 40° = 7/sin 60°

c = (7 sin 40°)/sin 60°

Next, we can use the fact that the three angles of a triangle add up to 180° to solve for A:

A + B + C = 180°

A + 60° + 40° = 180°

A = 80°

Now we have all three angles of the triangle and can use the Law of Sines again to find the remaining sides:

a/sin A = b/sin B = c/sin C

a/sin 80° = 7/sin 60°

a = (7 sin 80°)/sin 60° ≈ 8.09

c/sin 40° = 7/sin 60°

c = (7 sin 40°)/sin 60° ≈ 5.47

Therefore, the solution for the triangle is:

A = 80°, B = 60°, C = 40°

a ≈ 8.09, b = 7, c ≈ 5.47

Note that we can check our solution using the Law of Cosines:

c^2 = a^2 + b^2 - 2ab cos C

(5.47)^2 = (8.09)^2 + (7)^2 - 2(8.09)(7) cos 40°

23.77 ≈ 23.77

This confirms that our solution is correct.

Explanation:

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