Problem 60 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch each angle in standard position. (a) \(\frac{11 \pi}{6}\) (b) \(-\frac{2 \pi}{3}\)

Step-by-Step Solution

Answer

To sketch (a) \(\frac{11 \pi}{6}\), rotate \(60\) degrees or \(\frac{\pi}{6}\) radians from the positive x-axis in the clockwise direction. To sketch (b) \(-\frac{2\pi}{3}\), rotate \(120\) degrees or \(\frac{2\pi}{3}\) radians from the positive x-axis in the clockwise direction, which leads to an angle in the third quadrant.

Step 1: Sketch (a)

The initial side of each angle in standard position is the positive x-axis of a Cartesian plane. Next, we note that the given angle \( \frac{11 \pi}{6} \) is more than a full rotation of \(2\pi\) but less than \(2\pi\) radians. So, one way to find the terminal side of this angle is to subtract \(2\pi\) from it until you get an angle which lies between \(0\) and \(2\pi\). In this case, \(\frac{11 \pi}{6} - 2\pi = \frac{-\pi}{6}\). This is the same as rotating an angle of \(\frac{\pi}{6}\) in the clockwise direction. Therefore, to sketch it, you rotate about the origin from the positive x-axis \(60\) degrees or \(\frac{\pi}{6}\) radians clockwise.

Step 2: Sketch (b)

The angle is \(-\frac{2\pi}{3}\), a negative angle. A negative angle means we will have to rotate clockwise. The magnitude of the angle is larger than half a rotation \(\frac{2\pi}{3} > \frac{1\pi}{2}\), so the terminal side of this angle will lie in quadrants II or III. Specifically, \(-\frac{2\pi}{3}\) means we rotate \(120\) degrees or \(\frac{2\pi}{3}\) radians from the positive x-axis in the clockwise direction. This results in an angle in the third quadrant, terminating at a line which contains points \(({1/\sqrt{2}}, -{1/\sqrt{2}})\) and \((-{1/\sqrt{2}}, {1/\sqrt{2}})\).

Key Concepts

Angle Sketching

Angle sketching involves visually representing an angle on the coordinate plane. Here's how you go about it:

• The **initial side** of an angle is always positioned along the positive x-axis.
• An **angle** can be **positive** or **negative**, indicating a counterclockwise or clockwise rotation from the initial side respectively.
• The angle's endpoint, or the "terminal side," determines the direction and distance of rotation.

To sketch an angle, start by identifying its direction of rotation, then move accordingly from the initial side to mark the terminal side. Lastly, label the angle properly to complete the sketch.

Standard Position

An angle is said to be in the standard position when its initial side is on the positive x-axis and its vertex is at the origin of the coordinate plane. This positioning is essential for consistency and ease in measuring angles and defining trigonometric functions. Understanding the standard position helps us identify angles and distinguish them based on their quadrant affiliation:

• The **initial side** lies on the positive x-axis.
• The rotation happens about the origin to determine where the terminal side lands.
• An angle in the standard position allows for a uniform way to compare angles of different magnitudes.

Recognizing angles in their standard position simplifies calculations and enhances clarity when dealing with trigonometry.

Radian Measure

Radian measure is a method of measuring angles based on the radius of a circle. This system measures the actual distance traveled along the circle's circumference related to the radius. Here's a simple overview:

• A full circle is exactly **\(2\pi\) radians**, equivalent to **360 degrees**.
• You can express angles using radians as fractions of \(\pi\).
• To convert an angle from degrees to radians, use the formula: \(\text{{radians}} = \left(\frac{\pi}{180}\right) \times \text{{degrees}}\).

Radian measure allows for straightforward calculations in trigonometry as it integrates seamlessly with both equations and geometric interpretations. Understanding radians facilitates easier comprehension of angles as rotational measures.

Quadrants in Coordinate Plane

The coordinate plane is divided into four quadrants, each representing a unique combination of positive and negative values for x and y coordinates. Angles in standard position are grouped according to the quadrant their terminal sides fall into:

• **Quadrant I**: Both x and y coordinates are positive.
• **Quadrant II**: x is negative, y is positive.
• **Quadrant III**: Both x and y coordinates are negative.
• **Quadrant IV**: x is positive, y is negative.

Understanding the quadrants helps in recognizing where an angle's terminal side falls after rotation. Knowing the quadrant makes it easier to determine trigonometric values since the sign of the trig functions changes depending on the quadrant. This concept is vital for precise sketching and calculation of angles.