We don't know cos (105°) from any special triangle, but we do know cosine and sine for 60° and 45°. We hope that we don't have to spell out that cos (105°) = cos (60° + 45°). Let's plug our numbers into the cosine addition formula:
cos (60° + 45°) = cos (60°) cos (45°) – sin (60°) sin (45°)
Now we can use special triangles to find our values:
Example 2
Find the exact value of tan (-15°).
Nothing good can come from being so negative on the inside, tangent. We're talking plagues of locusts levels of bad here. You're an odd function, so spit some of that negativity on the outside.
tan ( -15°) = -tan 15°
The easiest way to break this up is by (30 – 45). Let's put it into our subtraction formula for tangent:
Watch the signs. We use positive 45 inside of tangent, but have subtraction in the numerator and addition in the denominator.
We are almost there, but could definitely use more some simplifying:
Those lonely little 3s can cancel each other out, giving us our answer.
Example 3
Simplify .
We can dive right in by using the subtraction formula for sine. Gotta go fast.
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