Solve the triangle ABC if angle A = 76 degrees and a = 12, b = 4
(Tan²θ * sin²θ) + cos²θ +2sin²θ = sec²θ
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To solve the triangle ABC, we need to find the values of the remaining side lengths and angles. In this case, we are given the length of side a (12), the length of side b (4), and angle A (76 degrees).
First, let's use the Law of Sines to find angle B. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant.
sin(A) / a = sin(B) / b
Substituting the given values:
sin(76) / 12 = sin(B) / 4
To find sin(B), we can rearrange the equation:
sin(B) = (sin(76) / 12) * 4
Now we can find the value of sin(B):
sin(B) = (sin(76) / 12) * 4 ≈ 0.8930
Next, we can use the inverse sine function (sin^(-1)) to find angle B:
B = sin^(-1)(0.8930) ≈ 63.24 degrees
Now that we know angle B, we can find angle C by subtracting angles A and B from 180 degrees:
C = 180 - A - B = 180 - 76 - 63.24 ≈ 40.76 degrees
Now, to find the remaining side lengths, we can use the Law of Sines again. Let's find side c:
sin(A) / a = sin(C) / c
Substituting the given values:
sin(76) / 12 = sin(40.76) / c
To find c, we can rearrange the equation:
c = (sin(40.76) / sin(76)) * 12
Calculating the value of c:
c = (sin(40.76) / sin(76)) * 12 ≈ 7.14
Therefore, the solved triangle ABC is:
Angle A = 76 degrees
Angle B ≈ 63.24 degrees
Angle C ≈ 40.76 degrees
Side a = 12
Side b = 4
Side c ≈ 7.14