Step by step
The question
Question: find cos 2θ when sin θ = 12/13, 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 = 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: x² = 25.
The square root of 25 is 5.
Plus or minus.
In quadrant I, x is positive: right of the y-axis.
So x = 5.
3 Find cos θ
Cos θ is x/r: 5/13.
So cos θ = 5/13.
4 Write the rule
The double-angle rule: cos 2θ = cos² θ − sin² θ.
Careful!
cos 2θ is not 2 times cos θ.
The 2 is inside: it doubles the angle, not the answer.
5 Work it out
Put in cos θ = 5/13 and sin θ = 12/13.
Brackets keep each one together.
Square each one: top times top, bottom times bottom.
5 squared is 25.
12 squared is 144.
Same bottom, 169: 25 minus 144 is -119.
So cos 2θ = −119/169.
6 Check
Let's check with the calculator.
The reference angle is tan⁻¹ of 12/5: about 67.38°.
In quadrant I, θ is about 67.38°.
Twice that is 134.76°.
cos of it is about −0.704.
And −119/169 is −0.704.
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.