sketch and solve a problem using Law of Sines and Law of Cosines with solutions.
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sketch and solve a problem using Law of Sines and Law of Cosines with solutions.
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Example problem that involves using both the Law of Sines and Law of Cosines to solve.
Problem:
A triangle has sides of length 5, 7, and 9. Find the measure of the largest angle in the triangle, to the nearest degree.
Solution:
Since we are given the lengths of all three sides of the triangle, we can use the Law of Cosines to find the measure of one of the angles. Let's use the angle opposite the side of length 9 and call it angle A:
cos(A) = (b^2 + c^2 - a^2) / (2bc)
where a = 9, b = 5, and c = 7
cos(A) = (5^2 + 7^2 - 9^2) / (2 * 5 * 7) = 0.4286
Taking the inverse cosine of both sides, we find:
A = cos^-1(0.4286) = 65.28 degrees (rounded to two decimal places)
Now that we know the measure of angle A, we can use the Law of Sines to find the measures of the other two angles. Let's use angle B as an example:
sin(B) / b = sin(A) / a
sin(B) / 5 = sin(65.28) / 9
sin(B) = (5 * sin(65.28)) / 9 = 0.6186
Taking the inverse sine of both sides, we find:
B = sin^-1(0.6186) = 38.75 degrees (rounded to two decimal places)
Finally, we can find the measure of angle C by subtracting the measures of angles A and B from 180 degrees:
C = 180 - A - B = 75.97 degrees (rounded to two decimal places)
Therefore, the largest angle in the triangle is angle C, which measures approximately 76 degrees.