Step by step
The question
Question: find cos 2θ when cos θ = −4/5, θ in quadrant II.
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 = −4 and r = 5.
r is a distance, always positive, so the minus belongs to x.
We draw it in quadrant II: a right triangle from O to the point.
2 Find the third side
Pythagoras: x² + y² = r².
−4 squared is 16.
5 squared is 25.
Take away 16 on both sides: y² = 9.
The square root of 9 is 3.
Plus or minus.
In quadrant II, y is positive: above the x-axis.
So y = 3.
3 Find sin θ
Sin θ is y/r: 3/5.
So sin θ = 3/5.
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.
−8/5 is not between −1 and 1, and a cosine always is.
5 Work it out
Put in cos θ = −4/5 and sin θ = 3/5.
Brackets keep each one together.
Square each one: top times top, bottom times bottom.
−4 squared is 16: the minus goes.
3 squared is 9.
Same bottom, 25: 16 minus 9 is 7.
So cos 2θ = 7/25.
6 Check
Let's check with the calculator.
The reference angle is tan⁻¹ of 3/4: about 36.87°.
In quadrant II, θ is about 143.13°.
Twice that is 286.26°.
cos of it is about 0.280.
And 7/25 is 0.280.
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.