Sin -270 degrees. Find the exact value. Please explain? 🙂
The Correct Answer and Explanation is:
To find the exact value of sin(−270°), we follow these steps:
Step 1: Use the unit circle
The sine of an angle is the y-coordinate of the point on the unit circle corresponding to that angle.
Step 2: Convert −270° to a positive angle
Since the unit circle is periodic every 360°, we can add 360° to −270°:−270∘+360∘=90∘-270^\circ + 360^\circ = 90^\circ−270∘+360∘=90∘
This means:sin(−270∘)=sin(90∘)\sin(-270^\circ) = \sin(90^\circ)sin(−270∘)=sin(90∘)
Step 3: Use known sine value
From the unit circle:sin(90∘)=1\sin(90^\circ) = 1sin(90∘)=1
Therefore:sin(−270∘)=1\sin(-270^\circ) = 1sin(−270∘)=1
Explanation (300 words)
To find the sine of −270 degrees, it is helpful to understand how negative angles behave and how the unit circle works. Angles measured in the counterclockwise direction are positive, while those measured clockwise are negative. So, −270 degrees is an angle measured 270 degrees in the clockwise direction from the positive x-axis.
On the unit circle, one full rotation is 360 degrees. This means that angles repeat every 360 degrees. When you subtract or add multiples of 360 degrees, the sine value does not change. This idea is called periodicity. Therefore, we can add 360 degrees to −270 degrees to find an equivalent angle:
−270 + 360 = 90 degrees.
Now we evaluate sine at 90 degrees. The point on the unit circle corresponding to 90 degrees lies on the positive y-axis, and its coordinates are (0, 1). The sine of an angle is the y-value of the point on the unit circle. Hence:sin(90∘)=1\sin(90^\circ) = 1sin(90∘)=1
And because we found that:sin(−270∘)=sin(90∘)\sin(-270^\circ) = \sin(90^\circ)sin(−270∘)=sin(90∘)
It follows that:sin(−270∘)=1\sin(-270^\circ) = 1sin(−270∘)=1
So, the exact value of sin(−270°) is 1. This method shows how periodicity and reference angles help simplify trigonometric expressions involving negative angles.
