Step by step
The question
Question: find cos 2θ when sin θ = 4/5, 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 = 4 and r = 5.
We draw it in quadrant I: 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: x² = 9.
The square root of 9 is 3.
Plus or minus.
In quadrant I, x is positive: right of the y-axis.
So x = 3.
3 Find cos θ
Cos θ is x/r: 3/5.
So cos 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.
6/5 is not between −1 and 1, and a cosine always is.
5 Work it out
Put in cos θ = 3/5 and sin θ = 4/5.
Brackets keep each one together.
Square each one: top times top, bottom times bottom.
3 squared is 9.
4 squared is 16.
Same bottom, 25: 9 minus 16 is -7.
So cos 2θ = −7/25.
6 Check
Let's check with the calculator.
The reference angle is tan⁻¹ of 4/3: about 53.13°.
In quadrant I, θ is about 53.13°.
Twice that is 106.26°.
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 cos θ.
Then: write the rule.
Then: work it out.
Last: check.