Final answer:
The range of a cosine function is the set of output values the function can take, which is always between -1 and +1. This applies to the standard cosine function and is unaffected by horizontal or phase shifts. This range is consistent with the definition of the cosine function as the ratio of the adjacent side to the hypotenuse in a right triangle.
Step-by-step explanation:
The range of a cosine function refers to the set of possible values that the function can output. In mathematical terms, the cosine function oscillates between +1 and -1 irrespective of any horizontal shifts or phase shifts. A horizontal shift, demonstrated in Figure 15.8 (b), where the function is shifted by an angle φ (phase shift), does not alter the range of values of the function, which remain – from its minimum value of -1 to its maximum of +1.
Similarly, in Figure 16.10, a sine function, which is related to the cosine function, also oscillates between +1 and -1 every 2π radians (a complete cycle). This oscillation represents the wave function amplitude, which in cases other than a cosine can fluctuate between +A and -A.
As illustrated in Figure 2.18, the cosine function can be visualized as the ratio of the adjacent side to the hypotenuse (Ax/A = cos A) in a right triangle, further highlighting that this ratio (and thus the range of the cosine function) is between -1 and 1.
For example, when √(1+1)* approaches 1, it is indicative that cos 0 = 1, representing one end of the cosine function's range.