Problem 18 - Contemporary Precalculus

Question

Determine whether or not the given angles in standard position are coterminal. $$170^{\circ},-550^{\circ}$$

Step-by-Step Solution

Answer

Answer: Yes, the angles $$170^{\circ}$$ and $$-550^{\circ}$$ are coterminal because we found an integer value for n (-2) such that $$170^{\circ} + 360^{\circ}(-2) = -550^{\circ}$$. This means both angles share the same terminal side in standard position.

Step 1: Write down the given angles and the coterminal angle equation

We are given two angles in standard position: $$170^{\circ}$$ and $$-550^{\circ}$$. We need to determine if they are coterminal. The general equation to check for coterminality is: $$A + 360^{\circ}n = B$$ where A is the first angle, B is the second angle, and n is an integer.

Step 2: Substitute the given angles into the coterminal angle equation

Now substitute the given angles into the equation: $$170^{\circ} + 360^{\circ}n = -550^{\circ}$$

Step 3: Solve for n

Rearrange the equation to solve for n: $$360^{\circ}n = -550^{\circ} - 170^{\circ}$$ $$360^{\circ}n = -720^{\circ}$$ Divide both sides by $$360^{\circ}$$ to get: $$n = \frac{-720^{\circ}}{360^{\circ}}$$ $$n = -2$$

Step 4: Determine if the angles are coterminal

Since we found an integer value for n (-2), the given angles are coterminal. In this case, when we add $$360^{\circ}$$ to $$170^{\circ}$$ two times, we end up with the same terminal side as $$-550^{\circ}$$: $$170^{\circ} + 360^{\circ}(-2) = -550^{\circ}$$

Key Concepts

Standard Position Angles

Imagine an angle where its vertex is at the origin of a coordinate system and one side, known as the initial side, is fixed along the positive x-axis. The other side, the terminal side, rotates to form the angle. This setup describes a standard position angle.

Angles measured in this way are easily compared, especially when determining coterminality; two angles are coterminal if their terminal sides coincide after completing a full rotation, regardless of the direction. For instance, an angle rotating clockwise might need to go around more than once to match the terminal side of one that rotates counter-clockwise.

Angle Measurement

Angles are commonly measured in degrees or radians, with each full rotation around a circle comprising $$360^{\text{degrees}}$$ or $$2\text{pi}$$ radians. Any angle can have an equivalent angle, known as a coterminal angle, that differs by full rotations from the original.

To find coterminal angles, you may add or subtract multiples of $$360^{\text{degrees}}$$ (or $$2\text{pi}$$ radians). For example, the angles $$170^{\text{degrees}}$$ and $$-550^{\text{degrees}}$$ may not look coterminal at first glance, but by understanding that angle measurement is cyclical, we can investigate their relationship.

Solving Equations

Solving equations is a foundational skill in mathematics, allowing us to find unknown values, known as variables. In our context, we solve for a variable that represents the number of full rotations needed to determine if angles are coterminal. This task is akin to solving a simple algebraic equation.

By arranging our equation, we isolate the variable—in this case, the integer $$n$$ which signifies full $$360^{\text{degrees}}$$ rotations—and calculate its value. When we obtain an integer for $$n$$, the angles in question are coterminal, as seen with the angles $$170^{\text{degrees}}$$ and $$-550^{\text{degrees}}$$, where the calculated $$n = -2$$ confirms their coterminality.