Show the solutions of the two examples given:
(a) If a = 4, b = 3, and A = 36o.
(b) If a = 3, b = 4, and A = 48.59o.
If we don't know what the questions are, how can we help you to find solutions?
A. 4*3 (given side lengths) = 12 then the area which is 36 divided by 12 = 3 which would = c or the other side length if that is what the question is asking.
B. 4*3=12 and 48.59/12=? use a calculator to figure that one out. sorry if this is wrong hope it helped:)
To find the solutions for these examples, we can use the laws of trigonometry, specifically the sine law and the cosine law. Let's break down each example separately:
(a) If a = 4, b = 3, and A = 36°:
1. Apply the sine law to find angle B:
The sine law states that a/sin(A) = b/sin(B), where A and B are angles opposite to sides a and b respectively.
Plugging in the given values, we have 4/sin(36°) = 3/sin(B).
Rearranging the equation, sin(B) = (3 * sin(36°)) / 4.
Using a calculator, we find sin(B) ≈ 0.69539.
Taking the inverse sine (sin^(-1)), we find B ≈ 43.47° or B ≈ 136.53°.
2. Now, to find angle C, we know that the sum of angles in a triangle is 180°.
C = 180° - A - B.
Substituting the known values, C = 180° - 36° - B ≈ 100.53° or C ≈ 43.47°.
Therefore, the solutions for Example (a) are:
A = 36°, B ≈ 43.47°, C ≈ 100.53°
or
A = 36°, B ≈ 136.53°, C ≈ 43.47°.
(b) If a = 3, b = 4, and A = 48.59°:
1. Apply the cosine law to find angle C:
The cosine law states that c^2 = a^2 + b^2 - 2ab * cos(C), where c is the side opposite to angle C.
Plugging in the given values, we have c^2 = 3^2 + 4^2 - 2(3)(4)cos(C).
Simplifying, we get c^2 = 25 - 24cos(C).
Using a calculator, we find cos(C) ≈ 0.96890.
Taking the inverse cosine (cos^(-1)), we find C ≈ 14.09° or C ≈ 345.91°.
2. To find angle B, we can use the sine law as in Example (a).
a/sin(A) = b/sin(B)
3/sin(48.59°) = 4/sin(B)
Rearranging the equation, sin(B) = (4 * sin(48.59°)) / 3.
Using a calculator, we find sin(B) ≈ 0.99731.
Taking the inverse sine (sin^(-1)), we find B ≈ 84.19° or B ≈ 95.81°.
Therefore, the solutions for Example (b) are:
A = 48.59°, B ≈ 84.19°, C ≈ 14.09°
or
A = 48.59°, B ≈ 95.81°, C ≈ 345.91°.
Remember to always check if the triangle actually exists by ensuring that the sum of the angles is 180°.