Question

Which of the following is not equal to sin⁡(-230°)?

sin⁡(130°)
-sin⁡(-50°)
sin⁡(50°)
sin⁡(-50°)

Answer 1

the answer is probably sin(50)

Answer 2

From trigonometry we know that:

if
`sin(\theta)=sin(\alpha)`

then,
`\theta=n\pi+(-1)^n\alpha` (where
`n` is an integer)

This can be rewritten in degrees as:

`\theta=n(180^(\circ))+(-1)^n\alpha`.............(Equation 1)

Now, in our case,
`\alpha=-230^(\circ)`

Therefore, (Equation 1) can be written as:

`\theta=n(180^(\circ))+(-1)^n(-230^(\circ))`..........(Equation 2)

Now, to find the correct options all that we have to do is replace n by relevant integers and find the values of
`\theta` that match.

For n=2, (Equation 2) gives us:
`\theta=2* 180^(\circ)+(-1)^2(-230^(\circ))=360^(\circ)-230^(\circ)=130^(\circ)`.

Thus,
`sin(230^(\circ))=sin(130^(\circ))`

Now, we know that:
`-sin(-50^(\circ))=sin(50^(\circ))`

Let n=-1, then:

`\theta=(-1)* 180^(\circ)+(-1)^(-1)(-230^(\circ))=-180^(\circ)+230^(\circ)=50^(\circ)`

Thus,
`sin(-230^(\circ))=-sin(-50^(\circ))`

Likewise,
`sin(-230^(\circ))=sin(50^(\circ))`

Only the last option
`sin(-50^(\circ))` will never match
`sin(-230^(\circ))` because no integral value of
`n` will ever give
`\theta=-50^(\circ)`

Thus the last option is the correct option.