Step by step
The question
Question: find cos 2θ when tan θ = −12/5, θ in quadrant IV.
We know tan θ, and the quadrant.
1 Draw the triangle
Take a point (x, y) at distance r from O.
tan θ is y over x.
So y = −12 and x = 5.
In quadrant IV, x is positive and y is negative: −12/5 is −12/5.
We draw it in quadrant IV: a right triangle from O to the point.
2 Find the third side
Pythagoras: r² = x² + y².
5 squared is 25.
−12 squared is 144.
Together 169.
r is the square root of 169: r = 13.
A distance, so positive.
3 Find sin θ and cos θ
Sin θ is y/r: −12/13.
So sin θ = −12/13.
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 IV, θ is about 292.62°.
Twice that is 585.24°.
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 and cos θ.
Then: write the rule.
Then: work it out.
Last: check.