Find the exact value of the sine, cosine, and tangent of the angle 330 degrees.
I know the tangent is -(square root 3)/3, but how?
330 ° = 360 ° - 30 °
Circular functions:
sin ( 360 ° - theta ) = - sin theta
sin 330 ° = sin ( 360 ° - 30 ° ) = - sin 30 ° = - 1 / 2
cos ( 360 ° - theta ) = cos theta
cos 330 ° = cos ( 360 ° - 30 ° ) = cos 30 ° = sqroot 2 / 2
tan ( 360 ° - theta ) = - tan theta
tan 330 ° = tan( 360 ° - 30 ° ) = - tan 30 ° = - sqroot 3 / 3
To find the exact value of the sine, cosine, and tangent of the angle 330 degrees, we can start by converting it to its equivalent angle within one full revolution, which is 330 degrees - 360 degrees = -30 degrees.
Since the unit circle repeats every 360 degrees, an angle of -30 degrees is equivalent to an angle of 330 degrees.
Next, we can use the unit circle to determine the values of the sine, cosine, and tangent for the angle of -30 degrees (or 330 degrees).
1. Sine (sin): Sine is the y-coordinate on the unit circle. For a -30 degrees angle (or 330 degrees), the y-coordinate is negative, and its absolute value is the same as the y-coordinate for an angle of 30 degrees.
So, the sine of 330 degrees is equal to the sine of 30 degrees, which is 1/2.
2. Cosine (cos): Cosine is the x-coordinate on the unit circle. For a -30 degrees angle (or 330 degrees), the x-coordinate is positive, and its absolute value is the same as the x-coordinate for an angle of 30 degrees.
So, the cosine of 330 degrees is equal to the cosine of 30 degrees, which is (√3)/2.
3. Tangent (tan): Tangent is the ratio of the sine to the cosine (tan = sin/cos). For the angle of 330 degrees, we can substitute the values we found for sine and cosine into this formula.
Tangent of 330 degrees = (Sine of 330 degrees) / (Cosine of 330 degrees)
= (1/2) / (√3)/2
= (1/2) * (2/(√3))
= 1/√3
= √3 / 3
= √3 / (-3) (since we are dealing with -30 degrees)
Therefore, the correct value for the tangent of the angle 330 degrees is -(√3)/3.