Problem 10 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 10 through 14, rewrite each of the following expressions in terms of a positive acute angle. This positive acute angle is sometimes referred to as a reference angle. (a) \(\sin \left(-48^{\circ}\right)\) (b) \(\cos \left(-48^{\circ}\right)\)

Step-by-Step Solution

Answer

(a) \(\sin \left(-48^{\circ}\right)\) can be rewritten as \(-\sin(48^{\circ})\). (b) \(\cos \left(-48^{\circ}\right)\) can be rewritten as \(\cos(48^{\circ})\).

Step 1: Convert Negative Angle to Positive

Both functions have a negative angle of -48. To convert it into a positive angle, add 360 degrees to get the positive equivalent: -48 + 360 = 312 degrees.

Step 2: Simplify sin(-48 degrees)

The sine function has a property that \(\sin(-x) = -\sin(x)\). So instead of \(\sin(-48^{\circ})\), we can replace it with \(-\sin(48^{\circ})\). The new expression is \(-\sin(48^{\circ})\).

Step 3: Simplify cos(-48 degrees)

The cosine function is even, so negative angles does not change the cosine function value. This means that \(\cos(-48^{\circ})\) is identical to \(\cos(48^{\circ})\). So, \(\cos(-48^{\circ})\) is rewritten as \(\cos(48^{\circ})\).

Key Concepts

Reference Angle

Imagine peeling away the complexity layer of an angle to its simplest, most understandable form. This is what a reference angle does. The reference angle is the smallest positive angle that you can draw from the terminal side of any angle to the x-axis. For trigonometric calculations, it is helpful as it allows us to work with acute angles, which are always between 0 and 90 degrees. Understanding the reference angle involves the following:

• Converting any given angle to a positive value by adding or subtracting full rotations (360 degrees for plane angles).
• Identifying the angle in its standard position and determining its acute reference angle.

This simplification is critical, because once you have the reference angle, you can determine the sine, cosine, and other trigonometric functions for angles beyond the first quadrant. For example, for a given angle of -48 degrees, this can be converted by adding 360 degrees to result in 312 degrees. The reference angle here would be 360 - 312 = 48 degrees, which is the angle related to our standard trigonometric circle.

Sine Function

The sine function measures the y-coordinate of the endpoint of an arc on the unit circle. Typically, it ranges between -1 and 1. An important aspect of the sine function is its behavior under negative angles, expressed by its odd function property:

• For any angle, \(\sin(-x) = -\sin(x)\). This indicates that the sine of a negative angle is the negative of the sine of the positive angle.
• This property helps convert complex expressions involving negative angles into simpler, more manageable positive expressions.

Returning to our example, under the transformation of the negative angle -48 degrees, we convert it to \(-\sin(48^{\circ})\). This illustrates how the sine function changes its sign depending on the quadrant in which the angle terminates, reinforcing its dependence on the direction.

Cosine Function

The cosine function, unlike the sine function, is an even function. This implies fascinating symmetry in the way it operates under negative angles. For any angle, \(\cos(-x) = \cos(x)\). In other words, the cosine of a negative angle is identical to that of the positive angle.

• This stability under sign change makes it easier to compute for negative angles.
• Particularly useful in practical scenarios where knowing the positive angle equivalent is sufficient for computations, simplifying the process.
• For instance, with our angle of -48 degrees, this property allows us to directly rewrite it as \(\cos(48^{\circ})\) without any additional steps needed.

This tells us that the value doesn't change with the direction the angle is measured, unlike the sine function. Thus, understanding cosine as an even function can simplify computations significantly when dealing with real-world applications or further trigonometric analyses.