Standardized Test Practice
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Graph the angle in standard position, find its reference angle, and determine the sign of the sine in the quadrant where the terminal side is located.
A
We want to find the exact value of sin 240^(∘). To do so, let's start by recalling some trigonometric values for special angles.
| 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 | csc 30^(∘)=2 | sec 30^(∘)=2sqrt(3)/3 | cot 30^(∘)=sqrt(3) |
| sin 45^(∘)=sqrt(2)/2 | cos 45^(∘)=sqrt(2)/2 | tan 45^(∘)=1 | csc 45^(∘)=sqrt(2) | sec 45^(∘)=sqrt(2) | cot 45^(∘)=1 |
| sin 60^(∘)=sqrt(3)/2 | cos 60^(∘)=1/2 | tan 60^(∘)=sqrt(3) | csc 60^(∘)=2sqrt(3)/3 | sec 60^(∘)=2 | cot 60^(∘)=sqrt(3)/3 |
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^(∘).
Next, we will recall the signs of the six trigonometric functions in the different quadrants of the coordinate plane.
In Quadrant III — the quadrant where the terminal side of the angle is located — sine is negative. With this information, we can write an equation relating the cosecant of the angle and the cosecant of its reference angle. sin 240^(∘) = - sin 60^(∘) Using our table, we can see that sin 60^(∘)= sqrt(3)2. sin 240^(∘) = - sin 60^(∘) ⇒ sin 240^(∘) =- sqrt(3)/2 This corresponds to option A.