Step by step
The question
Question: find sin 2θ when sin θ = 5/13, 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 = 5 and r = 13.
We draw it in quadrant I: a right triangle from O to the point.
2 Find the third side
Pythagoras: x² + y² = r².
5 squared is 25.
13 squared is 169.
Take away 25 on both sides: x² = 144.
The square root of 144 is 12.
Plus or minus.
In quadrant I, x is positive: right of the y-axis.
So x = 12.
3 Find cos θ
Cos θ is x/r: 12/13.
So cos θ = 12/13.
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.
5 Work it out
Put in sin θ = 5/13 and cos θ = 12/13.
Tops: 2 times 5 times 12 is 120.
Bottoms: 13 times 13 is 169.
So sin 2θ = 120/169.
6 Check
Let's check with the calculator.
The reference angle is tan⁻¹ of 5/12: about 22.62°.
In quadrant I, θ is about 22.62°.
Twice that is 45.24°.
sin of it is about 0.710.
And 120/169 is 0.710.
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.