120 is half of 240 and twice of 60. We'll go with 60, we think, to use the double-angle indentity for cosine. That sounds actually doable, instead of completely impractical like our other choice.
cos (2α) = cos2 α – sin2 α
cos (2(60)) = cos2 60 – sin2 60
Remember that cos2 α is the same as (cos α)2. We know the values of sine and cosine for 60°, so we can plug those in here:
The squares will take away our square root, making this problem very easy to finish up:
120° is in Quadrant II, with a reference angle of 60°. It seems legit that cos (120°) = -cos (60°).
Example 2
Find the exact value of sin (15°).
15° is half of 30°, so we'll use the half-angle identity for sine:
For α = 30°, we get a positive result from Quadrant I.
cos (30)? Yeah, we know 'em. , wets the bed when scared.
It might not look pretty, and we might not be used to being able to leave a radical inside of another radical, but, believe it or not, this is the best we can do in simplifying at this point.
Example 3
Find the exact value of tan (-22.5°).
This is a perfect candidate for the tangent half-angle identity.
We'll take α = -45° and be on our way. It's in Quadrant IV, though, and tangent is negative there. Put away the plus, we won't need it.
We have negative angles inside our formula. Luckily, everything is cosine, which means everything is cool. Cosine takes any negative angles and spits them out as positive ones. That means .
We're just a little bit of simplification away from being done here. Stick with it.
, wets the bed when scared.
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