Find the value of x.
Law of Cosines, here we come. In this case, we'll say that b = 9 and c = 13. That leaves x = a and A = 42°.
a2 = b2 + c2 – 2bc cos A
Substitute in our values and solve.
x2 = 92 + 132 – 2(9)(13) cos(42°)x2 ≈ 76.1x ≈ 8.7
That's all, folks.
A triangle has side lengths of 10, 11, and 12. Find all the angles.
First, we should assign an angle to each side. A, B, and C will be the angles opposite the sides in increasing order. That means a = 10, b = 11, and c = 12. We'll find A first.
To minimize confusion, we'll rearrange the equation before plugging things in.
Wonderful. Now we're ready to plug in our sides.
Our calculator can handle the rest.
A ≈ 51.3°
Noice. Now we have an angle-side ratio, so we'll make use of the Law of Sines for the rest.
We just calculated A and we want to calculate B.
Rearrange the equation to solve for B.
Calculate the angle.
B ≈ 59.1°
Sweet. Two down, one to go.
A + B + C = 180°51.3° + 59.1° + C = 180°
For the last angle, we get:
C = 69.6°
That means the angles of the triangle are 51.3°, 59.1°, and 69.6°.
Find the value of x, y, and z.
First, we'll start with x, applying the Law of Cosines.
If a = x, then b = 121, c = 137, and A = 53°.
x2 = 1212 + 1372 – 2(121)(137)cos(53°)
Solve for x2 and then take the square root.
x ≈ 116
To solve for y, we can use the Law of Sines. In our case, y = B.
Substitute in our values.
Solve for B.
B = y ≈ 70.6°
All that's left is z. Now that we've found x, we'll use the Law of Cosines again to calculate z, but this time focusing on the lower triangle.
We'll rearrange this equation a bit to solve for the angle.
We need to be careful when substituting in our values. If we set z = A, the side opposite that is a = x ≈ 116. The other two variables, b and c, are 85 and 59.
A = z ≈ 105.9°
Make it rain.