Problem 16 - Contemporary Precalculus

Question

Determine whether or not the given angles in standard position are coterminal. $$\frac{7 \pi}{6},-\frac{5 \pi}{6}$$

Step-by-Step Solution

Answer

Answer: Yes, the angles $$\frac{7 \pi}{6}$$ and $$-\frac{5 \pi}{6}$$ are coterminal.

Step 1: Find the common denominator

To compare the angles, first find a common denominator which, in this case, is 6. Convert both angles to fractions with a denominator of 6. $$\frac{7 \pi}{6}, -\frac{5 \pi}{6}$$

Step 2: Calculate the difference

Now, subtract one angle from the other to find the difference between them. $$\frac{7 \pi}{6} - \left(-\frac{5 \pi}{6}\right) = \frac{7 \pi}{6} + \frac{5 \pi}{6} = \frac{12 \pi}{6}$$

Step 3: Check if the difference is a multiple of $$2\pi$$

After finding the difference between the two angles, we need to check if the difference is a multiple of $$2\pi$$. $$\frac{12 \pi}{6} = 2\pi$$ The difference between the angles is exactly $$2\pi$$, which means that they are coterminal because they share the same terminal side.

Key Concepts

Radians

Radians are a unit of angular measurement used in mathematics and trigonometry. Unlike degrees, which divide a full circle into 360 parts, radians express angles in terms of the radius of a circle. One radian is the angle created when the arc length equals the radius of the circle. This results in a full circle being equal to \(2\pi\) radians.

Here's an easy way to convert between radians and degrees:

• To convert degrees to radians, multiply by \(\frac{\pi}{180}\).
• To convert radians to degrees, multiply by \(\frac{180}{\pi}\).

Radians are frequently used in calculus and higher-level mathematics because they offer a simpler relationship between angles and other mathematical functions. Understanding radians is crucial for solving problems involving trigonometric functions and for determining coterminal angles.

Angle Conversion

Angle conversion is the process of changing the measurement of an angle from one unit to another, such as from degrees to radians or vice versa. This is important in trigonometry and other fields of mathematics where different units might be used for different purposes.

To break it down:

• For converting degrees to radians, use the formula: \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\).
• For converting radians to degrees, use the formula: \(\text{degrees} = \text{radians} \times \frac{180}{\pi}\).

Understanding how to convert angles is essential when comparing angles or determining coterminal angles. Coterminal angles have the same standard position terminal side, so converting to a consistent unit ensures easy comparison and calculation.

Trigonometry Concepts

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It is built around the six main functions: sine, cosine, tangent, cosecant, secant, and cotangent. These functions are particularly helpful in understanding the properties of angles.

Coterminal angles are a key concept within trigonometry. These are angles that share the same terminal side when plotted in standard position (beginning at the positive x-axis). To find coterminal angles:

• Add or subtract \(2\pi\) radians (or \(360^\circ\)) to the given angle.
• Angles differing by multiples of \(2\pi\) are coterminal because they end at the same location on a circle.

Understanding coterminal angles is necessary for solving many trigonometric problems as they help in simplifying the periodic nature of trigonometric functions.