Determine the exact value of the trigonometric ratio below. Sketch the terminal arm on a Cartesian plane, determine the related acute angle, and show your work to receive full mark... Determine the exact value of the trigonometric ratio below. Sketch the terminal arm on a Cartesian plane, determine the related acute angle, and show your work to receive full marks.
Understand the Problem
The question is asking to determine the exact value of a trigonometric ratio and to sketch the terminal arm of an angle on the Cartesian plane. It also requests to find the related acute angle and show the work involved in obtaining the solution.
Answer
The exact value of $\sin\left(\frac{11\pi}{6}\right)$ is $-\frac{1}{2}$.
Answer for screen readers
The exact value of $\sin\left(\frac{11\pi}{6}\right)$ is $-\frac{1}{2}$.
Steps to Solve
- Convert the Angle to Radians
The given angle is $\frac{11\pi}{6}$. To find the equivalent degree measure, we can use: $$ \theta_{degrees} = \frac{11\pi}{6} \times \frac{180}{\pi} = 330^\circ $$
- Understanding the Location of the Angle
The angle $330^\circ$ is in the fourth quadrant. The reference angle (RAA) is calculated as: $$ RAA = 360^\circ - 330^\circ = 30^\circ $$
- Determine the Exact Value of the Sine Function
In the fourth quadrant, the sine value is negative. The sine of the reference angle $30^\circ$ is: $$ \sin(30^\circ) = \frac{1}{2} $$
Thus, the sine of $330^\circ$ is: $$ \sin(330^\circ) = -\frac{1}{2} $$
- Sketching the Terminal Arm
In the Cartesian plane, start by drawing the x and y axes. The angle $330^\circ$ starts from the positive x-axis and moves clockwise. Mark the terminal arm in the fourth quadrant.
- Determine the Related Acute Angle
The related acute angle is simply the RAA we calculated: $$ \text{Related Acute Angle} = 30^\circ $$
The exact value of $\sin\left(\frac{11\pi}{6}\right)$ is $-\frac{1}{2}$.
More Information
The terminal arm of the angle $\frac{11\pi}{6}$ in the fourth quadrant indicates that the sine function will yield a negative value. Additionally, the reference or related acute angle is $30^\circ$, which is commonly associated with a sine value of $\frac{1}{2}$.
Tips
- Confusing Quadrants: Always ensure to determine the correct quadrant for the angle and know the signs of trigonometric functions in each quadrant.
- Miscalculating the Reference Angle: Ensure to subtract from $360^\circ$ for angles in the fourth quadrant to get the reference angle.
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