Step by step
The question
Question: find sin 2θ when sin θ = −3/5, θ in quadrant III.
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.
r is a distance, always positive, so the minus belongs to y.
We draw it in quadrant III: 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 III, x is negative: left of the y-axis.
So x = −4.
3 Find cos θ
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 cos θ.
Then: write the rule.
Then: work it out.
Last: check.