Problem 58 - A Graphical Approach to Precalculus with Limits

Question

Begin by reproducing the graph in RiGuRE as. Keep in mind that for each of the four points labeled in the figure, \(r=1 .\) For each quadrantal angle, identify the appropriate values of \(x, y,\) and \(r\) to find the indicated function value. If it is undefined, say so. Check your answers with a calculator in degree mode. $$\sec \left(-540^{\circ}\right)$$

Step-by-Step Solution

Answer

\(\sec(-540^{\circ}) = -1\).

Step 1: Determine the Reference Angle

Start by finding the reference angle for \(-540^{\circ}\). We can find an equivalent angle between \(0^{\circ}\) and \(360^{\circ}\) by adding \(360^{\circ}\) repeatedly until the angle is positive. \(-540^{\circ} + 360^{\circ} \times 2 = \-540^{\circ} + 720^{\circ} = 180^{\circ}\).

Step 2: Identify the Quadrant

The angle \(180^{\circ}\) is on the negative x-axis in the Cartesian plane, which is the second quadrant. However, since it's exactly on the axis, it doesn't really belong to any quadrant, but the angle itself is past the first two quadrants.

Step 3: Extract x, y, and r Coordinates

For the point \((r, \theta)\) at angle \(180^{\circ}\), the corresponding Cartesian coordinates will be \((-1, 0)\) because the point lies on the negative x-axis with radius \(r = 1\). Thus, \(x = -1\), \(y = 0\), and \(r = 1\).

Step 4: Calculate sec(θ)

The secant function, \(\sec(\theta)\), is defined as \(\frac{r}{x}\). Substitute the values: \(\sec(180^{\circ}) = \frac{1}{-1} = -1\).

Step 5: Verify with a Calculator

Using a calculator in degree mode, input \(\sec(180^{\circ})\). Note that calculators typically provide the reciprocal function, so enter \(1/\cos(180^{\circ})\) to verify the result. This should also give \(-1\).

Key Concepts

Radians and Degrees

Understanding radians and degrees is essential in trigonometry as they are the standard units to measure angles. A full circle consists of 360 degrees or 2π radians. Converting between these units is straightforward, as the key is the relationship that

• 180 degrees = π radians, hence
• 1 degree = π/180 radians, and
• 1 radian = 180/π degrees.

To find equivalent angles, you can adjust your initial angle by multiples of 360 degrees (or 2π radians). For instance, if you have an angle like \(-540^{\circ}\), you could bring it between 0 and 360 degrees by adding 360 degrees several times, resulting in the equivalent \(180^{\circ}\).When using a calculator for trigonometric functions, ensure it is set to the correct angle mode, either degrees or radians, to avoid errors.

Reference Angles

A reference angle helps simplify the understanding of angles in various quadrants. It is the acute angle formed by the terminal side of the given angle and the x-axis. For angles measured in degrees:

• If an angle is between 0 and 360 degrees, the reference angle is simply itself if it's in the first quadrant.
• For angles in other quadrants, subtract it from the nearest multiple of 180 (for example, 180 or 360).

The reference angle for \(-540^{\circ}\) is \(180^{\circ}\) since after converting \(-540^{\circ}\) using the addition method, it falls exactly on an axis.Reference angles are particularly useful for evaluating trigonometric functions, as they can have the same sine, cosine, and secant values as their related angles.

Cartesian Coordinates

Cartesian coordinates are used to pinpoint locations in a plane using an ordered pair \((x, y)\). This system is vital for graphing functions and solving geometric problems. The key points in the plane are defined based on:

• The x-axis, which runs horizontally.
• The y-axis, running vertically.
• Origin, at the point \((0,0)\).

In trigonometric functions, the hypotenuse \(r\) plays a crucial role and represents the radius or the distance from the origin to the point.For \(180^{\circ}\), the Cartesian coordinates would be \((-1, 0)\), since the angle location is exactly on the negative x-axis with a radius \(r = 1\).Understanding these coordinates allows for the accurate calculation of trigonometric functions, including the secant.

Secant Function

The secant function (\(\sec\) ) is one of the six primary trigonometric functions, defined as the reciprocal of the cosine function. In terms of a right triangle, it is expressed as:\[\sec(\theta) = \frac{r}{x}\]Understanding secant involves:

• Recognizing that it measures the distance from the origin, relative to the x-coordinate.
• Realizing it can only be accurately determined where the x-coordinate is non-zero.

In the given case, \(\sec(-540^{\circ})\) or \(\sec(180^{\circ})\), you substitute the point \((-1, 0)\) with \(r = 1\) into the equation: \[\sec(180^{\circ}) = \frac{1}{-1} = -1\]Calculating secant involves converting angles to degrees or radians appropriately, and applying the formula, ensuring accuracy in solving trigonometric expressions. It is crucial to confirm results with a calculator, using reciprocal functions like \(1/\cos\).