Final answer:
To solve the triangle, we can use the Law of Cosines and the Law of Sines. By substituting the given values into the respective equations, we can find the length of side b, the measure of angle A, and the measure of angle C. The length of side b is approximately 88 m, the measure of angle A is approximately 36° 9', and the measure of angle C is approximately 77°.
Step-by-step explanation:
To solve the triangle, we can use the Law of Cosines. The Law of Cosines states that in a triangle with sides a, b, and c, and angle A opposite side a, the following relationship holds true: c^2 = a^2 + b^2 - 2ab*cos(A).
Given B = 66° 51', c = 36 m, and a = 78 m, we can substitute these values into the equation to find b. Rearranging the equation, we have b^2 = c^2 - a^2 + 2ab*cos(A). Plugging in the values, we get b^2 = 36^2 - 78^2 + 2(36)(78)*cos(66° 51'). Solving for b, we find that b ≈ 88 m.
To find angle A, we can use the Law of Sines. The Law of Sines states that in a triangle with sides a, b, and c, and angles A, B, and C, the following relationship holds true: a/sin(A) = b/sin(B) = c/sin(C).
Plugging in the values, we have 78/sin(A) = 88/sin(66° 51'). Solving for sin(A), we find sin(A) ≈ (78 * sin(66° 51')) / 88. Taking the inverse sine, we find A ≈ arcsin((78 * sin(66° 51')) / 88). Using a calculator, we find that A ≈ 36° 9'.
To find angle C, we can use the fact that the sum of the angles in a triangle is 180°. Therefore, C = 180° - A - B. Substituting the values, we find C ≈ 180° - 36° 9' - 66° 51'. Simplifying, we have C ≈ 77°.
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