Problem 11 - Contemporary Precalculus

Question

Find the radian measure of four angles in standard position that are coterminal with the angle in stan- dard position whose measure is given. $$\pi / 4$$

Step-by-Step Solution

Answer

Question: Find the radian measure of four angles in the standard position that are coterminal with $$\frac{\pi}{4}$$. Answer: The four angles in standard position that are coterminal with $$\frac{\pi}{4}$$ are: $$\frac{9\pi}{4}, \frac{17\pi}{4}, -\frac{7\pi}{4}, -\frac{15\pi}{4}$$.

Step 1: Find the first angle by adding $$2\pi$$

Add $$2\pi$$ to the given angle: $$\frac{\pi}{4} + 2\pi = \frac{\pi}{4} + \frac{8\pi}{4} = \frac{9\pi}{4}$$.

Step 2: Find the second angle by adding $$4\pi$$

Add $$4\pi$$ to the given angle: $$\frac{\pi}{4} + 4\pi = \frac{\pi}{4} + \frac{16\pi}{4} = \frac{17\pi}{4}$$.

Step 3: Find the third angle by subtracting $$2\pi$$

Subtract $$2\pi$$ from the given angle: $$\frac{\pi}{4} - 2\pi = \frac{\pi}{4} - \frac{8\pi}{4} = -\frac{7\pi}{4}$$.

Step 4: Find the fourth angle by subtracting $$4\pi$$

Subtract $$4\pi$$ from the given angle: $$\frac{\pi}{4} - 4\pi = \frac{\pi}{4} - \frac{16\pi}{4} = -\frac{15\pi}{4}$$. The four angles in standard position that are coterminal with $$\frac{\pi}{4}$$ are: $$\frac{9\pi}{4}, \frac{17\pi}{4}, -\frac{7\pi}{4}, -\frac{15\pi}{4}$$.

Key Concepts

Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position. This means they look the same when you draw them on a coordinate plane, although their measurements differ. To find coterminal angles, you can either add or subtract full rotations from a given angle.

• Full Rotation: In radians, a full rotation is represented by \(2\pi\).
• Adding Full Rotations: By adding \(2\pi\) to an angle, you get a positive coterminal angle.
• Subtracting Full Rotations: By subtracting \(2\pi\) (or more), you get a negative coterminal angle.

For instance, if you start with an angle of \(\pi/4\), adding \(2\pi\) gives you \(9\pi/4\), whereas subtracting \(2\pi\) results in \(-7\pi/4\). Both angles are coterminal with \(\pi/4\). This simple addition or subtraction allows us to explore a wide variety of angles all sharing the same position in a circle.

Standard Position

Angles in standard position start from the positive x-axis and move counterclockwise for positive angles. These angles help us consistently determine the precise location of an angle's terminal side.

• Vertex at Origin: The angle's vertex must be at the origin of the plane.
• Initial Side: The initial side of the angle lies along the positive x-axis.
• Counterclockwise for Positive Angles: Angles are measured counterclockwise when the angle is positive.

Why is standard position important? It creates a uniform way to describe angles, making them easy to compare and combine. This standard positioning enables a straightforward approach when dealing with situations like coterminal angles or calculations involving trigonometric functions.

Additive Property of Angles

The additive property of angles states that angles can be added together to form a new angle. This concept is essential when finding coterminal angles, as it involves adding or subtracting multiples of a full rotation.

• Combining Angles: You can always add two angles to make a bigger angle.
• Resultant Angle: Adding \(2\pi\) multiples to a smaller angle just repositions the terminal side, keeping the size different but visually identical in position.

Using this property, starting with an angle like \(\pi/4\), adding \(2\pi\) changes its measure to \(9\pi/4\), a different but coterminal angle. This demonstrates how adding full rotations doesn't change the angle's terminal side position, leveraging the additive property.

Angle Subtraction

Angle subtraction is a process where you reduce an angle's measure by a certain amount, such as \(2\pi\), to find another angle that is still coterminal. Subtraction works in much the same way as addition but in reverse.

• Creating Negative Angles: Subtracting \(2\pi\) (or more) from an angle can yield a negative angle.
• Coterminal Position: Despite subtraction, the angles will still share the same standard position.

For example, if you begin with \(\pi/4\) and subtract \(2\pi\), the outcome is \(-7\pi/4\). This means the angle \(-7\pi/4\) is a negative coterminal angle, situated similarly on the circle as \(\pi/4\). Angle subtraction thus allows you to explore angles beyond the range of a single circle rotation, revealing angles with identical geometric positions.