Step by step
The question
Question: find cos 2θ when cos θ = −12/13, θ in quadrant III.
We know cos θ, and the quadrant.
1 Draw the triangle
Take a point (x, y) at distance r from O.
cos θ is x over r.
So x = −12 and r = 13.
r is a distance, always positive, so the minus belongs to x.
We draw it in quadrant III: 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 III, y is negative: below the x-axis.
So y = −5.
3 Find sin θ
Sin θ is y/r: −5/13.
So sin 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.
−24/13 is not between −1 and 1, and a cosine always is.
5 Work it out
Put in cos θ = −12/13 and sin θ = −5/13.
Brackets keep each one together.
Square each one: top times top, bottom times bottom.
−12 squared is 144: the minus goes.
−5 squared is 25.
Same bottom, 169: 144 minus 25 is 119.
So cos 2θ = 119/169.
6 Check
Let's check with the calculator.
The reference angle is tan⁻¹ of 5/12: about 22.62°.
In quadrant III, θ is about 202.62°.
Twice that is 405.24°.
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 sin θ.
Then: write the rule.
Then: work it out.
Last: check.