In ∆ABC a = 7.8m, b=8.5m and B = 57.70 correct answer to 1d.p

1. Find angle A?
2. Find angle C?
3. Find angle c?

recall the law of sines

sinA/a = sinB/b
Once you have A, then C = 180-(A+B)

1. Angle A? Well, A stands for "Awesome!" It's the most amazing angle out there. But to find its numerical value, we can use the Law of Sines. So, get ready to flex those math muscles! Using the formula: sin(A) / a = sin(B) / b, we have the values a = 7.8m, b = 8.5m, and B = 57.70. Plug 'em all in and solve for angle A. Remember to round to 1 decimal place for maximum accuracy!

2. Angle C? Oh, angle C, the elusive one! We can find it using the interior angles of a triangle summing up to 180 degrees. Since we've already found angle A, we know that the sum of angles A and C is equal to 180 degrees minus angle B, which is 57.70 degrees. So, put on your math detective hat and solve for angle C by subtracting angle A from 180.

3. Angle c? Ah, angle c, the sidekick of angle C! To find its value, we can once again bring in the Law of Sines. Remember the formula: sin(C) / c = sin(A) / a. Plug in angle A, angle C (which we just solved), and the corresponding side lengths to calculate angle c.

To find the angles of triangle ABC, we can use the Law of Sines:

1. To find angle A:
The Law of Sines states that a/sin(A) = b/sin(B) = c/sin(C).

Given:
a = 7.8 m
b = 8.5 m
B = 57.70 degrees (corrected to 1 decimal place)

Using the formula, we can substitute the given values:
7.8 / sin(A) = 8.5 / sin(57.7)

Cross-multiplying:
7.8 * sin(57.7) = 8.5 * sin(A)

Dividing by 8.5:
(7.8 * sin(57.7)) / 8.5 = sin(A)

Using the inverse sine function:
A = sin^(-1)[(7.8 * sin(57.7)) / 8.5]

Calculating the value:
A ≈ 62.6 degrees (corrected to 1 decimal place)

Therefore, angle A is approximately 62.6 degrees.

2. To find angle C:
We know that the sum of the angles in a triangle is always 180 degrees.

Angle C can be calculated by subtracting angles A and B from 180 degrees:
C = 180 - A - B
C = 180 - 62.6 - 57.7

Calculating the value:
C ≈ 59.7 degrees (corrected to 1 decimal place)

Therefore, angle C is approximately 59.7 degrees.

3. To find angle c:
We can use the Law of Cosines to find angle c:

c^2 = a^2 + b^2 - 2ab * cos(C)

Given:
a = 7.8 m
b = 8.5 m
C = 59.7 degrees (corrected to 1 decimal place)

Substituting the values:
c^2 = 7.8^2 + 8.5^2 - 2 * 7.8 * 8.5 * cos(59.7)

Calculating the value:
c^2 ≈ 38.44

Taking the square root of both sides:
c ≈ √38.44
c ≈ 6.2 m (corrected to 1 decimal place)

Therefore, the length of side c is approximately 6.2 meters.

To find the answers to these questions, you can use the Law of Sines, which states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.

Let's solve the questions step by step:

1. Find angle A:
We have side a and angle B. We can use the Law of Sines as follows:
sin(A) / a = sin(B) / b
sin(A) / 7.8 = sin(57.70) / 8.5

First, convert the angle measurement to radians by multiplying by π/180:
sin(A) / 7.8 = sin(57.70 * π/180) / 8.5

Now, solve for sin(A) by multiplying both sides by 7.8:
sin(A) = (7.8 * sin(57.70 * π/180)) / 8.5

Lastly, find the angle A by taking the arcsine (inverse sine) of the value:
A = arcsin((7.8 * sin(57.70 * π/180)) / 8.5)

Calculate the value using your calculator, and round it to 1 decimal place to get the correct answer.

2. Find angle C:
We know angle A from the previous calculation. To find angle C, we can use the fact that the sum of angles in a triangle is equal to 180 degrees (π radians). Since we know angles A and B, we can calculate angle C:
C = 180 - A - B
C = 180 - Angle A - Angle B

Substitute the known values into the equation to find angle C. Remember to round the answer to 1 decimal place.

3. Find side c:
We have sides a, b, and angles A, B. To find side c, we can again use the Law of Sines:
sin(A) / a = sin(C) / c

Rearranging the equation to solve for c:
c = (a * sin(C)) / sin(A)
c = (7.8 * sin(C)) / sin(A)

Substitute the known values into the equation to find side c. Remember to round the answer to 1 decimal place.

That's how you can solve for angle A, angle C, and side c in triangle ABC using the given information and the Law of Sines.