Use the information to solve the triangle. If two solutions exist find both.
1.A=36°, B=98°, c=16
2.a=4, b=8, c=10
3.A=35°, b=8, c=12
To solve a triangle, you can use the law of sines and the law of cosines.
1. A=36°, B=98°, c=16:
To find the remaining side and angle, use the law of sines:
sin(A)/a = sin(B)/b = sin(C)/c
sin(36°)/a = sin(98°)/b = sin(C)/16
Solve for a:
sin(36°)/a = sin(98°)/16
a = sin(36°) * 16 / sin(98°)
a ≈ 9.91
Now, to find the remaining angle, use the fact that the sum of the angles in a triangle is 180°:
C = 180° - A - B
C = 180° - 36° - 98°
C ≈ 46°
So, the two solutions for this triangle are:
Solution 1: A = 36°, B = 98°, C ≈ 46°, a ≈ 9.91, b = 16, c = 16
Solution 2: A = 36°, B = 98°, C ≈ 134°, a = 16, b = 9.91, c = 16
2. a=4, b=8, c=10:
Given the lengths of the sides, we can use the law of cosines to find the angles:
c^2 = a^2 + b^2 - 2ab * cos(C)
Substituting the given values:
10^2 = 4^2 + 8^2 - 2 * 4 * 8 * cos(C)
100 = 16 + 64 - 64 * cos(C)
100 = 80 - 64 * cos(C)
64 * cos(C) = -20
cos(C) = -20 / 64
cos(C) ≈ -0.31
Now, we can use the inverse cosine function to find the value of angle C:
C = acos(cos(C))
C ≈ acos(-0.31)
C ≈ 110.27°
Since we know the angles A, B, and C, we can find the remaining side using the law of sines:
sin(A)/a = sin(B)/b = sin(C)/c
sin(110.27°)/a = sin(A)/4 = sin(B)/8 = sin(C)/10
Solve for a:
sin(110.27°)/a = sin(110.27°)/4
a = 4
So, there is only one solution for this triangle:
A ≈ 110.27°, B = 180° - A - C ≈ 180° - 110.27° - 110.27° ≈ -40.54° (Note: negative angle means it is outside the triangle), C ≈ 110.27°, a = 4, b = 8, c = 10
3. A=35°, b=8, c=12:
We can use the law of sines to find the remaining angles and side:
sin(A)/a = sin(B)/b = sin(C)/c
sin(35°)/a = sin(B)/8 = sin(C)/12
Solve for a:
sin(35°)/a = sin(B)/8
a = sin(35°) * 8 / sin(B)
a ≈ 5.13
Similarly, we can solve for angles B and C using the same equation:
sin(35°)/5.13 = sin(B)/8
sin(B) = sin(35°) * 8 / 5.13
B ≈ arcsin(sin(35°) * 8 / 5.13)
sin(C)/12 = sin(35°)/5.13
sin(C) = sin(35°) * 12 / 5.13
C ≈ arcsin(sin(35°) * 12 / 5.13)
So, the solution for this triangle is:
A = 35°, B ≈ arcsin(sin(35°) * 8 / 5.13), C ≈ arcsin(sin(35°) * 12 / 5.13), a ≈ 5.13, b = 8, c = 12
Note: The solutions for angles B and C may have multiple values, as the arcsin function has a range of -90° to 90°.