Problem 51 - Precalculus with Calculus Previews

Question

In Problems \(47-52,\) find the angle between 0 and \(2 \pi\) that is coterminal with the given angle. $$ -4 $$

Step-by-Step Solution

Answer

The coterminal angle is approximately 2.28 radians.

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides but may have different rotations. We find a coterminal angle by adding or subtracting multiples of \(2\pi\) (the full circle) until the angle is positive and less than \(2\pi\).

Step 2: Convert -4 to a Positive Angle

Since \(-4\) radians is negative, we need to add \(2\pi\) repeatedly until we obtain an angle between \(0\) and \(2\pi\).

Step 3: Calculate the Coterminal Angle

Start with \(-4\), and add \(2\pi \) (approximately \(6.28\)) to \(-4\): \(-4 + 2\pi \approx -4 + 6.28 = 2.28\). This angle \(2.28\) radians is between \(0\) and \(2\pi\) radians.

Step 4: Verify the Coterminal Angle

Check that \(2.28\) radians is a valid coterminal angle, as it falls between \(0\) and \(2\pi\) radians. This confirms that \(2.28\) is the correct coterminal angle.

Key Concepts

Radian Measure

Radian measure is a way to express angles using the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians focus on the arc length compared to the circle's radius.
In essence, a full circle measures exactly \(2\pi\) radians, which is equivalent to 360 degrees. Why \(2\pi\)? Because the circumference of a circle is \(2\pi\) times the radius (\(C = 2\pi r\)), and in radian terms, we consider the ratio of the arc length to the radius.
When you have an angle in radians, it directly tells you how many 'radius lengths' the arc of the angle would span.

• An angle of \(\pi\) radians represents a half-circle, or 180 degrees.
• \(\frac{\pi}{2}\) radians means you've moved through a quarter-circle, or 90 degrees.

Understanding radian measure can make calculations involving angular velocity, rotations, and trigonometric functions more intuitive and mathematically elegant.

Angle Conversion

Converting between radians and degrees is a common task in trigonometry. To convert radians to degrees, you use the conversion factor \(\frac{180}{\pi}\). Conversely, to go from degrees to radians, multiply by \(\frac{\pi}{180}\).
For example:

• To convert \(\pi\) radians into degrees: Multiply \(\pi \times \frac{180}{\pi} = 180\) degrees.
• Converting 45 degrees into radians: Multiply \(45 \times \frac{\pi}{180} = \frac{\pi}{4}\) radians.

Angle conversion becomes particularly useful when solving problems involving coterminal angles. When given an angle in degrees, you can convert it to radians to utilize trigonometric functions more effectively, especially in calculus or complex numbers.

Trigonometric Functions

Trigonometric functions are essential in connecting geometry with algebra. They define relationships in right triangles and the unit circle.
Using radians, trigonometric functions take on a natural expression of periodicity:

• The sine and cosine functions repeat every \(2\pi\) radians, mirroring a full circle’s rotation.
• Tangent has a period of \(\pi\), as it reflects the slope’s repetition every half-circle.
• Functions such as secant and cosecant represent the reciprocal relationships with cosine and sine, respectively, while cotangent is the inverse of tangent.

These functions are also defined by their unique properties on the unit circle:

• For any angle \( \theta \), \(\sin(\theta)\) represents the y-coordinate, while \(\cos(\theta)\) is the x-coordinate, meaning at \(\theta = 0\), \(\cos(0) = 1\) and \(\sin(0) = 0\).
• The trigonometric identities like \(\sin^2(\theta) + \cos^2(\theta) = 1\) provide valuable insights for simplifying expressions.

With knowledge of trigonometric functions, solving equations and understanding rotations and harmonic motion becomes significantly more intuitive.