Step by step
The question
Question: find tan 2θ when sin θ = 3/5, θ in quadrant I.
We know sin θ, and the quadrant.
1 Draw the triangle
Take a point (x, y) at distance r from O.
sin θ is y over r.
So y = 3 and r = 5.
We draw it in quadrant I: a right triangle from O to the point.
2 Find the third side
Pythagoras: x² + y² = r².
3 squared is 9.
5 squared is 25.
Take away 9 on both sides: x² = 16.
The square root of 16 is 4.
Plus or minus.
In quadrant I, x is positive: right of the y-axis.
So x = 4.
3 Find tan θ
Tan θ is y/x: 3/4.
So tan θ = 3/4.
4 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.
5 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.
6 Check
Let's check with the calculator.
The reference angle is tan⁻¹ of 3/4: about 36.87°.
In quadrant I, θ is about 36.87°.
Twice that is 73.74°.
tan of it is about 3.429.
And 24/7 is 3.429.
The same!
So it is right.
Let's look back.
First: draw the triangle.
Then: find the third side.
Then: find tan θ.
Then: write the rule.
Then: work it out.
Last: check.