Step by step
The question
Question: find tan 2θ when tan θ = 5/12, θ in quadrant III.
We know tan θ, and the quadrant.
1 Write the rule
The double-angle rule: tan 2θ = 2 tan θ over (1 − tan² θ).
Careful!
tan 2θ is not 2 times tan θ.
The 2 is inside: it doubles the angle, not the answer.
2 Work it out
Put in tan θ = 5/12.
On top: 2 times 5/12 is 5/6.
At the bottom: 5/12 squared is 25/144.
1 is 144/144.
So 1 minus 25/144 is 119/144.
A fraction over a fraction.
Dividing by a fraction is times its flip: times 144/119.
Multiply and make it simpler: tan 2θ = 120/119.
3 Check
Let's check with the calculator.
The reference angle is tan⁻¹ of 5/12: about 22.62°.
In quadrant III, θ is about 202.62°.
Twice that is 405.24°.
tan of it is about 1.008.
And 120/119 is 1.008.
The same!
So it is right.
Let's look back.
First: write the rule.
Then: work it out.
Last: check.