Problem 59 - Precalculus with Calculus Previews

Question

Discuss how it is possible to determine without the aid of a calculator that both \(\sin 4\) and \(\cos 4\) are negative.

Step-by-Step Solution

Answer

Both \(\sin 4\) and \(\cos 4\) are negative because \(4\) radians place it in the third quadrant.

Step 1: Understanding the Angle in the Unit Circle

The given angles are in radians. Since we know that the full rotation in a circle in radians is \(2\pi\), when we determine if \(4\) radians falls on a specific quadrant, we first find how many radians \(4\) is equivalent to in terms of the full circle. We find that \(4 \approx 1.273\) rotations since \(\frac{4}{2\pi} \approx 0.6366\). This means \(4\) radians is in the second rotation past \(2\pi\).

Step 2: Finding the Quadrant of the Angle

Next, we will find the equivalent angle of \(4\) radians in terms of \(2\pi\). Subtract \(2\pi\) from \(4\):\[ 4 - 2\pi \approx 4 - 6.2832 = -2.2832 \] which is adjusted by another \(\pi\) to fall within \([0, 2\pi]\), giving us an equivalent angle of approximately \(5.999\) (since adding \(\pi\) rotates it into the quadrant needed) which edges into the third quadrant between \(\pi\) and \(3\pi/2\).

Step 3: Determining the Sign of Sine and Cosine in the Third Quadrant

In the unit circle, in the third quadrant, both sine and cosine values are negative. This is because, in this quadrant, the coordinates of the point corresponding to an angle are in the form \((-x, -y)\), where both values are negative.

Key Concepts

Radians

Understanding radians is an essential step for dealing with angles on the unit circle. A radian is a way to measure angles based on the radius of a circle. By definition, a full circle is equal to \(2\pi\) radians. This is because the circumference of a circle (the distance around it) is \(2\pi\) times its radius. Thus, the radian directly relates to the arc length of the circle.

When you're working with radians and angles, it's important to grasp how these angles translate into rotations around the unit circle. For instance, if you have an angle like \(4\) radians, you'll need to see how many times this angle "goes around" the circle. As mentioned, \(4\) radians equals about \(0.6366\) of a full rotation, or more intuitively, it's a rotation that goes beyond a single circle (past \(2\pi\)), yet doesn't complete a second full circle.

This understanding of radians helps in determining the quadrant an angle like \(4\) radians is located in, which leads us into the next crucial concept, the quadrants.

Quadrants

The unit circle is divided into four quadrants. These quadrants help us understand where an angle lies, which is crucial for determining the signs of trigonometric functions such as sine and cosine. Each quadrant represents a quarter of a complete revolution (or circle), and each has unique properties concerning the signs of sine and cosine.

• Quadrant I: All sine and cosine values are positive. This quadrant encompasses angles from \(0\) to \(\pi/2\).
• Quadrant II: Sine is positive, cosine is negative. Angles here range from \(\pi/2\) to \(\pi\).
• Quadrant III: Both sine and cosine are negative. This quadrant covers angles from \(\pi\) to \(3\pi/2\).
• Quadrant IV: Sine is negative, cosine is positive. It includes angles from \(3\pi/2\) to \(2\pi\).

When dealing with \(4\) radians, we need to convert it to determine which quadrant it falls into. As explored in the step-by-step solution, \(4\) radians simplifies to being just shy of \(6\) radians — specifically around \(5.999\), positioning it clearly within the range of the third quadrant. In this quadrant, both the x and y coordinates are negative, which leads us to the signs of the angles' sine and cosine values.

Sine and Cosine Signs

The sine and cosine functions represent the y-coordinate and x-coordinate of a point on the unit circle, respectively. Depending on which quadrant an angle falls into, the signs of these functions change.

In the third quadrant, both the sine and cosine values are negative. This happens because in the unit circle, every point has coordinates \(-x, -y\). The x-value, which corresponds to cosine, and the y-value, which corresponds to sine, are both negative in this quadrant. This is why for \(4\) radians, both the sine and cosine turn out to be negative numbers.

To summarize:

• Sine is the vertical (y-axis) coordinate.
• Cosine is the horizontal (x-axis) coordinate.
• In Quadrant III, both these values are negative as both x and y coordinates are below the origin or center of the circle.

Understanding these signs based on quadrants simplifies determining without computation whether trigonometric functions are positive or negative. This insight allows one to quickly evaluate trigonometric expressions by visualizing or calculating an angle's location in the unit circle.