find the minimum and maximum value of sin100 degree plus cos100 degree
To find the minimum and maximum values of \( \sin(100^\circ) + \cos(100^\circ) \), we can start by considering the range of values that both sine and cosine can take.
The sine function has a range of \([-1, 1]\), meaning that its output can vary from -1 to 1. Similarly, the cosine function also has a range of \([-1, 1]\).
Since we are adding the sine and cosine of the same angle (100 degrees), their sum will be highest when both sine and cosine have their maximum values, and it will be lowest when they have their minimum values.
Therefore, the maximum value of \( \sin(100^\circ) + \cos(100^\circ) \) will be obtained when both sine and cosine are at their maximum, which is \( \sin(90^\circ) + \cos(0^\circ) \). We know that \( \sin(90^\circ) = 1 \) and that \( \cos(0^\circ) = 1 \). Thus, the maximum value is \( 1 + 1 = 2 \).
On the other hand, the minimum value of \( \sin(100^\circ) + \cos(100^\circ) \) will be obtained when both sine and cosine are at their minimum, which is \( \sin(270^\circ) + \cos(180^\circ) \). We know that \( \sin(270^\circ) = -1 \) and that \( \cos(180^\circ) = -1 \). Hence, the minimum value is \( -1 + (-1) = -2 \).
Therefore, the minimum value of \( \sin(100^\circ) + \cos(100^\circ) \) is -2, and the maximum value is 2.