Problem 50 - Precalculus: A Concise Course

Question

Find the reference angle \(\boldsymbol{\theta}^{\prime}\), and sketch \(\boldsymbol{\theta}\) and \(\boldsymbol{\theta}^{\prime}\) in standard position. $$ \theta=\frac{7 \pi}{6} $$

Step-by-Step Solution

Answer

The reference angle \(\boldsymbol{\theta}^{\prime}\) is \(\frac{\pi}{6}\).

Step 1: Identify the Quadrant and Corresponding Angle

The given angle \(\frac{7 \pi}{6}\) is more than \(\pi\) and less than \(\frac{3\pi}{2}\), which means it's in the third quadrant. The corresponding angle for third quadrant is \(\pi\).

Step 2: Calculate the Reference Angle

The reference angle \(\boldsymbol{\theta}^{\prime}\) can be calculated by subtracting the corresponding angle of the quadrant from the given angle \(\boldsymbol{\theta}\). So, \(\boldsymbol{\theta}^{\prime}\) = \( |\frac{7 \pi}{6} - \pi| = \frac{\pi}{6}\).

Step 3: Drawing the Angle and Reference Angle

Draw a circle representing a unit circle. Draw the angle \(\boldsymbol{\theta}\) starting from the positive x-axis to the third quadrant making an angle of \(\frac{7 \pi}{6}\) with the x-axis. After that, draw the reference angle \(\boldsymbol{\theta}^{\prime}\) in the positive direction to the x-axis. Its magnitude will be of \(\frac{\pi}{6}\), which position in first quadrant.

Key Concepts

Unit Circle

The Unit Circle is an essential tool for understanding angles and trigonometry. It's a circle with a radius of one, centered at the origin of a coordinate system. This circle helps in visualizing angles and their trigonometric values.

• On the unit circle, the angle is measured beginning from the positive x-axis, moving counterclockwise.
• It allows easy determination of sine and cosine values which are the y-coordinate and x-coordinate of the point on the circle, respectively.

The unit circle simplifies complex angle problems and is fundamental for understanding rotational symmetries. It offers an intuitive understanding of how angles relate to coordinates and distances on the plane.

Standard Position

Understanding angles in standard position is crucial as it sets a consistent starting point. An angle is said to be in standard position when its vertex is at the origin and its initial side lies along the positive x-axis.

• The movement away from the initial side is counterclockwise for positive angles and clockwise for negative angles.
• This setup helps in determining the position of angles on the unit circle and forming reference angles.

Visualizing angles from this standard start lowers complexity by keeping the reference consistent, thus aiding in comparing angles and easily calculating associated trigonometric values.

Third Quadrant

The third quadrant of the coordinate system is found between \(\pi\) and \(\frac{3\pi}{2}\). When dealing with angles in this quadrant, certain properties help in better understanding and calculation.

• Both the sine and cosine values are negative in this quadrant.
• This quadrant actively affects the calculation of reference angles as seen in our exercise.
• To find the reference angle in the third quadrant, we subtract \(\pi\) from the given angle.

By comprehending the specific properties of the third quadrant, calculations involving trigonometric angles become much easier and more intuitive.

Trigonometric Angles

Trigonometric angles are fundamental to understanding circular movement and relationships on the unit circle.

• They are measured in radians or degrees, where \(2\pi\) radians represent a full circle.
• Key angles such as \(0, \frac{\pi}{2}, \pi,\) and \(\frac{3\pi}{2}\) define the quadrant boundaries.
• The reference angle is the smallest angle that an angle makes with the x-axis, instrumental in simplification.

Knowing how trigonometric angles operate within these boundaries aids in calculating exact values for sine, cosine, and other trigonometric functions, providing a deeper grasp of the relationships between angles and their coordinates.