Step by step
The question
Question: find sin 2θ when tan θ = 3/4, θ in quadrant III.
We know tan θ, and the quadrant.
1 Draw the triangle
Take a point (x, y) at distance r from O.
tan θ is y over x.
So y = −3 and x = −4.
In quadrant III, x is negative and y is negative: −3 over −4 is 3/4.
We draw it in quadrant III: a right triangle from O to the point.
2 Find the third side
Pythagoras: r² = x² + y².
−4 squared is 16.
−3 squared is 9.
Together 25.
r is the square root of 25: r = 5.
A distance, so positive.
3 Find sin θ and cos θ
Sin θ is y/r: −3/5.
So sin θ = −3/5.
Cos θ is x/r: −4/5.
So cos θ = −4/5.
4 Write the rule
The double-angle rule: sin 2θ = 2 sin θ cos θ.
Careful!
sin 2θ is not 2 times sin θ.
The 2 is inside: it doubles the angle, not the answer.
−6/5 is not between −1 and 1, and a sine always is.
5 Work it out
Put in sin θ = −3/5 and cos θ = −4/5.
Tops: 2 times 3 times 4 is 24.
Bottoms: 5 times 5 is 25.
Two minus signs make plus.
So sin 2θ = 24/25.
6 Check
Let's check with the calculator.
The reference angle is tan⁻¹ of 3/4: about 36.87°.
In quadrant III, θ is about 216.87°.
Twice that is 433.74°.
sin of it is about 0.960.
And 24/25 is 0.960.
The same!
So it is right.
Let's look back.
First: draw the triangle.
Then: find the third side.
Then: find sin θ and cos θ.
Then: write the rule.
Then: work it out.
Last: check.