To find the exact value of the real number y in the equation y = y \cdot \csc(11^\circ), we first need to understand the relationship between y and the cosecant function.
Cosecant is the reciprocal of the sine function, so we can rewrite the equation as follows:
y = \frac{y}{\sin(11^\circ)}
To simplify this equation, we can multiply both sides by \sin(11^\circ) (as long as \sin(11^\circ) \neq 0, which it isn’t, since 11 degrees is not an angle that yields a sine of zero):
y \sin(11^\circ) = y
Now, we can rearrange the equation:
y \sin(11^\circ) – y = 0
Factoring out y, we get:
y (\sin(11^\circ) – 1) = 0
This gives us two possibilities:
- y = 0
- \sin(11^\circ) – 1 = 0 (This is not possible since \sin(11^\circ) is not equal to 1)
Therefore, the only solution is:
y = 0
In conclusion, the exact value of y in the equation y = y \csc(11^\circ) is:
0