Question

Multiple Choice Which of the following is not in the range of the sine function? (a) \(\frac{\pi}{4}\) (b) \(\frac{3}{2}\) (c) \(-0.37\) (d) \(-1\)

Short answer

(b) \(\frac{3}{2}\)

Step-by-step solution

- Understanding the Range of the Sine Function

The sine function, denoted as \( \sin(x) \), outputs values that are always between \( -1 \) and \( 1 \) inclusive. Therefore, the range of the sine function is \[ -1 \leq \sin(x) \leq \ 1 \].

- Evaluating Each Option

Evaluate whether each option falls within the range identified in Step 1. \ewline \(a) \frac{\pi}{4} \approx 0.785 \) which is in the range \[ -1, 1 \]. \ewline \(b) \frac{\ 3}{2} = 1.5 \), which is outside of the range \[ -1, 1 \]. \ewline \(c) \-0.37 \), which is in the range \[ -1, 1 \]. \ewline \(d) \-1 \), which is in the range \[ -1, 1 \].

Key concepts

trigonometric functions

Trigonometric functions are fundamental in mathematics, especially in the study of triangles and oscillatory phenomena. These functions include sine, cosine, and tangent, among others. Each trigonometric function correlates an angle to a specific ratio of sides in a right-angled triangle.

For instance:

• The sine function relates the angle to the ratio of the length of the opposite side to the hypotenuse.
• The cosine function relates the angle to the ratio of the adjacent side to the hypotenuse.
• The tangent function is the ratio of the opposite side to the adjacent side.

Trigonometric functions are crucial in various fields such as physics, engineering, and even finance due to their periodic nature, which helps analyze cycles and patterns.

sine function

The sine function, denoted as \( \sin(x) \), is one of the most important trigonometric functions. It calculates the y-coordinate of a point on the unit circle corresponding to a given angle measured from the positive x-axis. The sine function is periodic and oscillates between -1 and 1.

Some key properties of \( \sin(x) \):

• The amplitude of the sine function is 1, meaning its maximum and minimum values are 1 and -1, respectively.
• It has a period of \( 2\pi \) because it repeats its values every \( 2\pi \) radians.
• \( \sin(x) \) is an odd function, which implies \( \sin(-x) = -\sin(x) \).

Understanding these properties helps in solving various problems and recognizing patterns in physics and engineering.

range of functions

The range of a function is the set of all possible output values. For the sine function, the range is all real numbers between -1 and 1 inclusive.

In mathematical notation, this is written as: \[ -1 \leq \sin(x) \leq 1 \] This means that for any real input \( x \), the sine function will always produce an output between -1 and 1.

When solving for whether a number is in the range of a sine function, check if it lies between -1 and 1. For example, \( \frac{3}{2} \) or 1.5 does not belong to this range. By understanding the range, you will be better equipped to solve equations and understand the behavior of the sine function in various applications.