Question

How do you solve a triangle given angle A = 18.27°, angle C = 134.13°, side A = 8.3?

Answer 1

`color(red)(B = 27.6)`

`color(red)(c ~~ 19.003)` (rounded to nearest thousandth's place)

`color(red)(b ~~ 12.266)` (rounded to nearest thousandth's place)

Explanation:

First, we know that the angles of a triangle add up to `180^@`. We have two angles (`A` and `C`), so we can find B by doing:
`180^@ - 18.27^@ - 134.13^@ = 27.6`

Therefore, angle `color(red)(B = 27.6)`.

Now, use the Law of Sines to solve for side `c`:

`c/sinC = a/sinA`

`c/sin134.13^@ = 8.3/sin18.27^@`

`c = (8.3sin134.13)/sin18.27^@`

Therefore,
side `color(red)(c ~~ 19.003)`

Since we have two sides and an angle (or more), we can use the Law of Cosines to solve for the last side, `b`.

The Law of Cosines is `b = sqrt(a^2 + c^2 - 2(a)(c)(cosB)`

Let's plug in our values into the formula:
`b = sqrt((8.3)^2 + (19.003)^2 - 2(8.3)(19.003)(cos27.6^@))`

`b = sqrt(68.89 + 361.114 - 279.553)`

`b = sqrt(430.004 - 279.553)`

`b = sqrt(150.451)`

Therefore,
`color(red)(b ~~ 12.266)` (rounded to nearest thousandth's place)

Hope this helps!