Step by step
The question
Question: find tan 2θ when tan θ = −3/4, θ in quadrant II.
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 = −3/4.
On top: 2 times −3/4 is −3/2.
At the bottom: −3/4 squared is 9/16.
1 is 16/16.
So 1 minus 9/16 is 7/16.
A fraction over a fraction.
Dividing by a fraction is times its flip: times 16/7.
Multiply and make it simpler: tan 2θ = −24/7.
3 Check
Let's check with the calculator.
The reference angle is tan⁻¹ of 3/4: about 36.87°.
In quadrant II, θ is about 143.13°.
Twice that is 286.26°.
tan of it is about −3.429.
And −24/7 is −3.429.
The same!
So it is right.
Let's look back.
First: write the rule.
Then: work it out.
Last: check.