Problem 86 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine two coterminal angles in radian measure (one positive and one negative) for each angle. (There are many correct answers). (a) \(-\frac{7 \pi}{8}\) (b) \(\frac{\pi}{12}\)

Step-by-Step Solution

Answer

The positive and negative coterminal angles for -7π/8 are 9π/8 and -23π/8 respectively. And for π/12, they are 25π/12 and -23π/12 respectively.

Step 1: Determine the Coterminal Angles of -7π/8

You can find a positive and a negative coterminal angle by adding and subtracting 2π to/from the given angle respectively.\nPositive Coterminal Angle of -7π/8: \[ -\frac{7 \pi}{8} + 2\pi = -\frac{7 \pi}{8} + \frac{16 \pi}{8} = \frac{9 \pi}{8} \]\nNegative Coterminal Angle of -7π/8: \[ -\frac{7 \pi}{8} - 2\pi = -\frac{7 \pi}{8} - \frac{16 \pi}{8} = -\frac{23 \pi}{8} \]

Step 2: Determine the Coterminal Angles of π/12

You can find a positive and a negative coterminal angle by adding and subtracting 2π to/from the given angle respectively.\nPositive Coterminal Angle of π/12: \[ \frac{\pi}{12} + 2\pi = \frac{\pi}{12} + \frac{24 \pi}{12} = \frac{25 \pi}{12} \]\nNegative Coterminal Angle of π/12: \[ \frac{\pi}{12} - 2\pi = \frac{\pi}{12} - \frac{24 \pi}{12} = -\frac{23 \pi}{12} \]

Key Concepts

Radian Measure

Radian measure is a way of expressing the size of an angle by comparing the length of an arc it intercepts to the radius of the circle. In contrast to degrees, which divide a circle into 360 equal parts, radians use the radius of the circle as the basic unit.

The circumference of a circle is given by the formula \( 2\pi r \) where \( r \) is the radius. Since the circumference represents the arc length of an angle that has gone 'full circle', or 360 degrees, it correlates to \( 2\pi \) radians.

Therefore, a radian is approximately \( 360 \/ (2\pi) \) degrees, or roughly 57.3 degrees. When we say that an angle measures 1 radian, we mean that the length of the arc, intercepted by that angle on a unit circle where \( r = 1 \) is exactly the same as the length of the radius.

Positive and Negative Angles

Angles can be classified as positive or negative based on the direction of their rotation from the start point. A positive angle is generated by counter-clockwise rotation from the initial side to the terminal side, which is the standard direction for measuring angles. On the other hand, a negative angle involves clockwise rotation.

To find coterminal angles, we can add or subtract any multiple of \( 2\pi \) radians (or 360 degrees, for degree measure) to the given angle. This is because adding \( 2\pi \) radians corresponds to making one full circle around the unit circle. For example, if you have an angle of \( -\frac{7 \pi}{8} \) radians and you add \( 2\pi \) radians, the rotation goes one full circle beyond the original negative angle, landing you at a positive angle that shares the same terminal side.

Angle Addition and Subtraction

The principles of angle addition and subtraction are fundamental in finding coterminal angles, as well as in many other trigonometric applications. Adding angles corresponds to a rotational movement where the rotation of one angle continues where the other left off. Subtracting an angle is like reversing the rotational movement.

When solving for coterminal angles, we utilize angle addition or subtraction by adding or subtracting whole multiples of \( 2\pi \) to the given angle. This does not change the terminal position of the angle, but it does alter its measure, reflecting the number of full rotations made around the circle. As an educational tip, visualizing this movement on the unit circle can help students understand the periodic nature of the angles and trigonometric functions.