Solving SSA Triangles

Example 1

In this triangle we know

• angle B = 31°
• b = 8
• and c = 13

 

In this case, we can use The Law of Sines first to find angle C:

sinC/c = sinB/b

sinC/13 = sin(31°)/8

So sinC = (13×sin(31°))/8

So C = sin^-1(0.8369...)

Next, we can use 'angles of a triangle' to find angle A:

A = 180° - 31° - 56.818...°

Now we can use The Law of Sines again to find a:

a/sinA = b/sinB

So a/sin(92.181...°) = 8/sin(31°)

Did you notice that we didn't use A = 92.2°. That angle is rounded to 1 decimal place. It's much better to use the unrounded number 92.181...° which you should still have on your calculator screen from the last calculation.

So a = (sin(92.181...°) × 8)/sin(31°)

So, we have completely solved the triangle ...

... or have we?

The other possible angle is:

C = 180° - 56.818...°

With a new value for C we should also re-calculate angle A and side a

Use 'angles of a triangle add to 180°' to find angle A:

A = 180° - 31° - 123.181...°

Now we can use The Law of Sines again to find a:

a/sinA = b/sinB

So a/sin25.818...° = 8/sin31°

So a = (sin25.818...°×8)/sin31°

Example 2

This is also an SSA triangle.

In this triangle we know angle angle M = 125°, m = 12.4 and l = 7.6

We will use The Law of Sines to find angle L first:

sinL/l = sinM/m

sinL/7.6 = sin125°/12.4

So sinL = (7.6×sin125°)/12.4

So L = 30.136...°

Next, we will use 'angles of a triangle' to find angle N:

N = 180° - 125° - 30.136...°

Now we will use The Law of Sines again to find n:

n/sinN = m/sinM

So n/sin24.863...° = 12.4/sin125°

So n = (sin24.863...°×12.4)/sin125°