Problem 43 - A Graphical Approach to Precalculus with Limits

Question

Refer to Example \(4 .\) Do not use a calculator. Let s correspond to the point ( \(x, y\) ) on the unit circle. (a) Determine the quadrant where ( \(x, y\) ) is located. (b) Determine whether sin \(s\) and cos \(s\) are positive or negative. $$s=-\frac{3 \pi}{4}$$

Step-by-Step Solution

Answer

Quadrant III; both sine and cosine are negative.

Step 1: Identify the Reference Angle

First, we find the reference angle for \( s = -\frac{3\pi}{4} \). Note that on the unit circle, negative angles are measured clockwise from the positive x-axis. Since \( -\frac{3\pi}{4} \) is negative, we add \( 2\pi \) to convert into a positive angle that falls within the standard range: \( -\frac{3\pi}{4} + 2\pi = \frac{5\pi}{4} \). This is the reference angle in radians.

Step 2: Determine the Quadrant

Next, determine which quadrant \( \frac{5\pi}{4} \) lies in. When an angle is given in radians, recall that: \(0 \leq x < \frac{\pi}{2}\) implies Quadrant I, \(\frac{\pi}{2} \leq x < \pi\) implies Quadrant II, \(\pi \leq x < \frac{3\pi}{2}\) implies Quadrant III, and \(\frac{3\pi}{2} \leq x < 2\pi\) implies Quadrant IV. \(\frac{5\pi}{4} \approx 3.93\) radians, which places it in Quadrant III (as \(\pi = 3.14\) and \(\frac{3\pi}{2} = 4.71\)).

Step 3: Evaluate Signs of Sine and Cosine

In Quadrant III, both sine and cosine are negative. Thus, at the point corresponding to \( -\frac{3\pi}{4} \), both \( \sin(s) \) and \( \cos(s) \) are negative.

Step 4: Conclusion

For point \( s = -\frac{3\pi}{4} \) on the unit circle, located in Quadrant III, \( \sin(s) \) and \( \cos(s) \) are both negative.

Key Concepts

Quadrants

The unit circle is divided into four quadrants, each representing a range of angles. Understanding these quadrants helps us determine the signs of trigonometric functions like sine and cosine.

• Quadrant I: Angles range from 0 to \(\frac{\pi}{2}\) (90 degrees). In this quadrant, both \(\sin\) and \(\cos\) are positive.
• Quadrant II: Angles range from \(\frac{\pi}{2}\) to \(\pi\) (180 degrees). Here, \(\sin\) is positive, while \(\cos\) is negative.
• Quadrant III: Angles range from \(\pi\) to \(\frac{3\pi}{2}\) (270 degrees). In this case, both \(\sin\) and \(\cos\) are negative.
• Quadrant IV: Angles range from \(\frac{3\pi}{2}\) to \(2\pi\) (360 degrees). \(\sin\) is negative, whereas \(\cos\) is positive.

So, if we have an angle like \(s = -\frac{3\pi}{4}\), converting it to a positive angle helps us locate it. For \(\frac{5\pi}{4}\), it lands us in Quadrant III.

Reference Angle

A reference angle is the acute angle that a given angle makes with the nearest x-axis. It's always measured positively, and is typically less than \(\frac{\pi}{2}\) or 90 degrees. This concept is crucial because it allows us to simplify calculations by reducing complex angles to one of the standard angles like \(30^{\circ}, 45^{\circ},\) or \(60^{\circ} \).

• For angles measured clockwise (negative angles), it's useful to convert them into positive equivalents by adding \(2\pi\) or 360 degrees.
• This aligns them into the unit circle's first cycle, simplifying calculations.

For \(s = -\frac{3\pi}{4}\), converting by adding \(2\pi\) results in \(\frac{5\pi}{4}\). The reference angle then is \(\frac{5\pi}{4} - \pi = \frac{\pi}{4}\), a standard angle.

Signs of Sine and Cosine

The signs of \(\sin\) and \(\cos\) depend on the quadrant where the angle is located. This is important in determining the exact values of trigonometric functions.

• In Quadrant I, both sine and cosine are positive because the point (x, y) is located in the top-right of the unit circle where both x and y coordinates are positive.
• In Quadrant II, the sine remains positive (as y is positive), but the cosine turns negative because x is negative.
• Quadrant III sees both sine and cosine as negative since x and y are both negative here.
• Finally, in Quadrant IV, sine is negative because y is negative, while cosine is positive due to x being positive.

When \(s = -\frac{3\pi}{4}\), its equivalent in Quadrant III means both the sine and cosine are negative.

Radians

Radians are a unit of measuring angles based on the radius of a circle. While degrees divide a circle into 360 parts, radians divide it into \(2\pi\) parts.
This makes calculating angles in trigonometry more natural and easier to work with mathematically.

• One complete revolution of a circle in radians is \(2\pi \), equivalent to 360 degrees.
• A right angle is \(\frac{\pi}{2}\), equivalent to 90 degrees.
• Radians provide a seamless connection between angle measures and arc lengths.

In the exercise, the angle \(s = -\frac{3\pi}{4}\) is originally given in radians. This facilitates the direct application of trigonometric identities and functions to find the quadrant and signs for sine and cosine.