Step by step
The question
Question: find sin 2x when cos x = 12/13, x in quadrant I.
We know cos x, and the quadrant.
1 Draw the triangle
Take a point (x, y) at distance r from O.
cos x is x over r.
So x = 12 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.
12 squared is 144.
13 squared is 169.
Take away 144 on both sides: y² = 25.
The square root of 25 is 5.
Plus or minus.
In quadrant I, y is positive: above the x-axis.
So y = 5.
3 Find sin x
Sin x is y/r: 5/13.
So sin x = 5/13.
4 Write the rule
The double-angle rule: sin 2x = 2 sin x cos x.
Careful!
sin 2x is not 2 times sin x.
The 2 is inside: it doubles the angle, not the answer.
5 Work it out
Put in sin x = 5/13 and cos x = 12/13.
Tops: 2 times 5 times 12 is 120.
Bottoms: 13 times 13 is 169.
So sin 2x = 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, x 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 sin x.
Then: write the rule.
Then: work it out.
Last: check.