Use the Law of Sines to find the missing angle of the triangle. Find m∠B given that c = 83, a = 44, and
m∠A = 31.
A. 76.3°
B. 15.8°
C. 72.7°
D. 164.2°
Need help on how I would go about doing this...step-by-step. Thank you!
the law of sines says that
sinC/c = sinA/a
sinC/83 = sin31°/44
sinC = 0.9715
So, C = 76.3°
A+B+C = 180, so
B = 180 - (31+76.3) = 72.7°
To solve for the missing angle in a triangle using the Law of Sines, you will use the equation:
sin(A) / a = sin(B) / b = sin(C) / c
where A, B, and C are the angles of the triangle, and a, b, and c are the lengths of the opposite sides respectively.
Given that angle A is 31°, side a is 44, and side c is 83, you want to solve for angle B.
1. Write down the Law of Sines equation, substituting the known values:
sin(31) / 44 = sin(B) / 83
2. Cross multiply the equation:
44 * sin(B) = 83 * sin(31)
3. Simplify the equation:
sin(B) = (83 * sin(31)) / 44
4. Use the inverse sine function (sin^(-1)) to find the value of B:
B = sin^(-1)((83 * sin(31)) / 44)
5. Calculate B using a calculator:
B ≈ 15.8°
Therefore, the correct answer is B. The missing angle m∠B is approximately 15.8°.
To use the Law of Sines to find the missing angle of a triangle, follow these steps:
Step 1: Identify the given information:
- Side a, which has a length of 44 units
- Side c, which has a length of 83 units
- Angle A, which has a measure of 31 degrees
Step 2: Recall the Law of Sines: 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. In formula form, this can be written as:
a / sin(A) = c / sin(C) = b / sin(B)
Step 3: Rearrange the Law of Sines to solve for angle B:
Since we are trying to solve for angle B, rearrange the formula to isolate sin(B):
sin(B) = (b / a) * sin(A)
Step 4: Plug in the known values:
In this case, the known values are:
- a = 44
- c = 83
- A = 31
So the formula becomes:
sin(B) = (b / 44) * sin(31)
Step 5: Solve for sin(B):
Calculate the value of sin(B) by plugging the values into the formula.
sin(B) = (b / 44) * sin(31)
Step 6: Rearrange the formula to solve for angle B:
To isolate B, take the inverse sine of both sides of the equation:
B = arcsin((b / 44) * sin(31))
Step 7: Calculate the missing angle:
Plug in the known values to calculate the missing angle B.
Using a calculator:
B ≈ arcsin((b / 44) * sin(31))
B ≈ arcsin((b / 44) * 0.515)
B ≈ arcsin(0.0117 * b)
B ≈ arcsin(0.0117 * 83)
B ≈ arcsin(0.9702)
B ≈ 72.7°
Therefore, the missing angle B is approximately 72.7 degrees.
So, the correct answer is C. 72.7°.