Problem 52 - Basic Technical Mathematics with Calculus

Question

The given angles are in standard position. Designate each angle by the quadrant in which the terminal side lies, or as a quadrantal angle. $$12 \mathrm{rad},-2 \mathrm{rad}$$

Step-by-Step Solution

Answer

12 radians is in the fourth quadrant; -2 radians is in the third quadrant.

Step 1: Understanding the Problem

We need to determine the position of two angles given in radians: \(12\) radians and \(-2\) radians. Specifically, we are to identify the quadrants where their terminal sides lie or determine if they are quadrantal angles (angles that lie on the x-axis or y-axis).

Step 2: Converting Radians to Angles in a Circle

The angles are in radians, and since every full rotation around a circle is \(2\pi \) radians or approximately \(6.283\) radians, angles larger than \(2\pi\) radians must have equivalent smaller angles between \(0\) and \(2\pi\). Similarly, negative angles can be converted to positive angles by adding \(2\pi\) radians.

Step 3: Angle for 12 Radians

To find the equivalent angle of \(12\) radians in a standard circle, we take the modulus with \(2\pi\) radians:\[12 mod 2\pi \approx 12 mod 6.283 \approx 5.434\text{ radians}\].This equivalent positive angle is \(5.434\text{ radians}\).

Step 4: Determining Quadrant for 12 Radians

Since \(5.434\text{ radians} < 2\pi\text{ radians},\) but \(5.434\text{ radians} > \frac{3\pi}{2}\text{ radians} \approx 4.712\text{ radians},\) the angle falls in the fourth quadrant.

Step 5: Angle for -2 Radians

For the angle \(-2\) radians, we convert it to a positive angle by adding \(2\pi\text{ radians}:\)\[-2 + 2\pi \approx -2 + 6.283 \approx 4.283\text{ radians}.\]So, the equivalent angle is \(4.283\text{ radians}.\)

Step 6: Determining Quadrant for -2 Radians

Since \(4.283\text{ radians} > \pi \approx 3.142\text{ radians},\) but \(4.283\text{ radians} < \frac{3\pi}{2}\approx 4.712\text{ radians},\) the angle falls in the third quadrant.

Key Concepts

Radian Measure

Radian measure is a way of expressing angles based on the radius of a circle.
One radian is the angle created when the arc length is equal to the radius of the circle. This is crucial because it provides a natural, math-friendly way of measuring angles.

• The full circle is represented by an angle of \( 2\pi \) radians, or approximately 6.283 radians.
• Often, radians are used in calculus and geometry because of the simpler integration and differentiation they allow.
• Radians help simplify many mathematical expressions compared to degrees.

It's important to understand that when working with trigonometry problems or when analyzing circular motion, radians can offer a clearer insight into the nature of the angles involved.

Angle Conversion

Sometimes, it's necessary to convert angles from radians to degrees and vice versa. This is because some problems or calculations may be easier in one form than the other. To convert an angle from radians to degrees, you can use the formula: \[ \text{Degrees} = \text{Radians} \times \left( \frac{180^{\circ}}{\pi} \right) \] Conversely, to convert degrees to radians, use: \[ \text{Radians} = \text{Degrees} \times \left( \frac{\pi}{180^{\circ}} \right) \]

• These formulas can be helpful when graphing angles and solving trigonometric equations.
• The process of converting allows you to work in your preferred unit, ensuring clearer understanding and computation.

Remember, whether in radians or degrees, angle measurement is integral to solving many types of geometry and trigonometry problems.

Quadrants

The concept of quadrants is fundamental in trigonometry and helps determine the position of angles.
In a plane divided by the x-axis and y-axis, each quadrant represents a unique region.

• Quadrant I: Both x and y coordinates are positive. This is where angles between 0 and \( \frac{\pi}{2} \) radians reside.
• Quadrant II: x-coordinates are negative, y-coordinates are positive. Angles between \( \frac{\pi}{2} \) and \( \pi \) radians can be found here.
• Quadrant III: Both x and y coordinates are negative. For angles that range from \( \pi \) to \( \frac{3\pi}{2} \).
• Quadrant IV: x-coordinates are positive, but y-coordinates are negative. These angles fall between \( \frac{3\pi}{2} \) and \( 2\pi \) radians.

Knowing the quadrant helps predict the sign of the trigonometric functions like sine, cosine, and tangent. It also assists in quickly finding angle measures based on the given radian or degree value.

Standard Position Angles

Angles in standard position have their vertex at the origin of the coordinate system and their initial side along the positive x-axis.
This setup is crucial for consistency in angle measurement.

• Because they start from the positive x-axis, angles in standard position offer a clear frame of reference.
• This format assists in determining the quadrant in which the terminal side, or the side that moves, lies.
• Standard position helps in visualizing angular movements across the coordinate plane, making it easier to predict outcomes of various trigonometric functions.

Understanding standard position angles is key to grasping the basics of trigonometry, as they provide a foundational block for solving more complex angle records and trigonometric equations.