range of
sin(siny) +cos(siny)
sin y can take on values between -1 and 1.
For any value of siny between -1 and 1, the range of
sin(siny) + cosine(siny) is between
-sqrt2 and +sqrt2.
Extreme values of sin(siny) + cos(siny) occur where sin(siny) = cos(siny)
siny = +or- (sqrt2)/2
To find the range of the function f(x) = sin(sin(y)) + cos(sin(y)), we need to determine the set of possible values that f(x) can take.
Since sin and cos functions have ranges between -1 and 1, we need to examine the range of their inputs, sin(y) and cos(sin(y)).
The range of sin(y) is [-1, 1]. Therefore, the range of cos(sin(y)) will also be bounded between -1 and 1, as the cosine function is defined on the entire real number line.
Now, by adding sin(sin(y)) and cos(sin(y)), we obtain the expression f(x) = sin(sin(y)) + cos(sin(y)). Since both terms are bounded between -1 and 1, the sum of these terms will also be bounded between -2 and 2.
Therefore, the range of the function f(x) = sin(sin(y)) + cos(sin(y)) is [-2, 2].