We want to find the exact value of tan 240. To do so, let's start by recalling some trigonometric values for special angles in the first quadrant.
Trigonometric Values for Special Angles
Sine
Cosine
Tangent
Cosecant
Secant
Cotangent
sin 30^(∘)=1/2
cos 30^(∘)=sqrt(3)/2
tan 30^(∘)=sqrt(3)/3
cot 30^(∘)=sqrt(3)
csc 30^(∘)=2
sec 30^(∘)=2sqrt(3)/3
sin 45^(∘)=sqrt(2)/2
cos 45^(∘)=sqrt(2)/2
tan 45^(∘)=1
cot 45^(∘)=1
csc 45^(∘)=sqrt(2)
sec 45^(∘)=sqrt(2)
sin 60^(∘)=sqrt(3)/2
cos 60^(∘)=1/2
tan 60^(∘)=sqrt(3)
cot 60^(∘)=sqrt(3)/3
csc 60^(∘)=2sqrt(3)/3
sec 60^(∘)=2
Next, let's graph θ =240^(∘) in standard position so that we can find its reference angle. This way we can use the values from our table.
Note that the terminal side of this angle lies in Quadrant III. Therefore, to find its reference angle θ' we will subtract 180^(∘) from 240^(∘).
In Quadrant III — the quadrant where the terminal side of the angle is located — tangent is positive. With this information, we can write an equation relating the tangent of the angle and the tangent of its reference angle.
tan 240^(∘) = tan 60^(∘)
Using our table, we can see that tan 60^(∘)= sqrt(3).
tan 240^(∘) = tan 60^(∘) ⇒
tan 240^(∘) = sqrt(3)
Therefore, the answer is option D.
