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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

you have to use the law of sines which a/sinA=b/sin B=c/ sin C where the top letter is the opposite side of the angle and the lower letter is the angle itself. first you would go about solving this for angle B by putting 12/sin 76=4/ sin B. doing simple algebra the equation comes out to (4*sin(76))/12=sin B then you take the inverse sin of both sides to get angle B. Once you have angle B all the inside angle of a triangle equal 180 degrees so solve for angle C by taking 180-A-B=C. Then you go about solving for side c using the same equation to find Angle B except it would come out to look like (a*sin(C))/sin(A)=c. You want to use side a and angle A because those are given to you in the equation and will give the best result. Hope this helped!