Question

How do i solve this trigonometry question? (Section 7.3)

Answer 1

Solve:
`tan(-(\pi)/(12))`

Now, a tangent function is an odd function, and you can find the proof on the internet, so we can say that:

`f(-x) = -f(x)`

`tan(-x) = -tan(x)`

`tan(-(\pi)/(12)) = -tan((\pi)/(12))`

Now, we just need to find what
`tan((\pi)/(12))` is.
Since
`(\pi)/(12)` is not an exact angle, we need to manipulate it in some way that it will be yield exact values.


`(\pi)/(3) - (\pi)/(4) = (4\pi - 3\pi)/(12) = (\pi)/(12)`

Thus, we can say:

`-tan((\pi)/(12)) = -tan((\pi)/(3) - (\pi)/(4))`

Using difference formula to simplify the expression, we know that the difference formula states:

`tan(A - B) = (tanA - tanB)/(1 + tanA \cdot tanB)`

`-tan((\pi)/(3) - (\pi)/(4)) = -(tan((\pi)/(3) - tan((\pi)/(4)))/(1 + tan((\pi)/(3)) \cdot tan((\pi)/(4)))`

The exact value for
`tan((\pi)/(3)) = √(3)`
The exact value for
`tan((\pi)/(4)) = 1`

Thus:

`-(tan((\pi)/(3) - tan((\pi)/(4)))/(1 + tan((\pi)/(3)) \cdot tan((\pi)/(4))) = -(√(3) - 1)/(1 + √(3))`

`= (1 - √(3))/(1 + √(3))`

Rationalise the denominator:

`(1 - √(3))/(1 + √(3)) \cdot (1 - √(3))/(1 - √(3))`

`= ((1 - √(3))^(2))/(1 - 3)`

Expand the perfect square:

`((1 - √(3))^(2))/(1 - 3) = (1 - 2√(3) + 3)/(-2)`

`= (4 - 2√(3))/(-2)`

`= -2 + √(3)`


`\therefore  <span>tan(-(\pi)/(12)) = √(3) - 2</span>`