Question

How do you use the double angle or half angle formulas to simplify tan 300?

Answer 1

Find tan 300.

Ans: `sqrt3`

Explanation:

`tan 300 = sin 300/(cos 300)`. First find sin 300 and cos 300.
cos 600 = cos (240 + 360) = cos 240 = cos (60 + 180) =
= - cos 60 = `- 1/2`.
Apply the 2 trig identities:
`cos 2x = 1 - 2sin^2 x` and `cos 2x = 2cos^2 x - 1`
`cos 600 = - 1/2 = 2cos^2 (300) - 1`
`2cos^2 (300) = 1 - 1/2 = 1/2` -->`cos^2 (300) = 1/4.`
`cos (300) = +- 1/2`. Since cos 300 is negative, then
`cos 300 = -1/2`
`cos (600) = -1/2 = 1 - 2sin^2 (300)`
`2sin^2 (300) = 1 + 1/2 = 3/2` --> `sin^2 (300) = 3/4`
`sin (300) = +- sqrt3/2`. Since sin (300) is negative then,
`sin (300) = -sqrt3/2`
`tan 300 = sin 300/(cos 300) = (-sqrt3/2)(-2/1) = sqrt3`

Check by calculator.
`tan 300 = 1.73 = sqrt3`.OK