Find each exact function value. See Example 2. sin 7π/6

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the exact value of \( \sin \left( \frac{7\pi}{6} \right) \).
2. Step 2: Recognize that \( \frac{7\pi}{6} \) is an angle in radians. Convert it to degrees if necessary for better understanding: \( \frac{7\pi}{6} \times \frac{180}{\pi} = 210^\circ \).
3. Step 3: Determine the reference angle. Since \( 210^\circ \) is in the third quadrant, the reference angle is \( 210^\circ - 180^\circ = 30^\circ \).
4. Step 4: Recall that in the third quadrant, the sine function is negative. The sine of the reference angle \( 30^\circ \) is \( \frac{1}{2} \).
5. Step 5: Apply the sign from the third quadrant to the reference angle's sine value: \( \sin \left( \frac{7\pi}{6} \right) = -\frac{1}{2} \).

Key Concepts

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it allows for the definition of sine, cosine, and tangent functions based on the coordinates of points on the circle. For any angle θ, the sine value corresponds to the y-coordinate and the cosine value corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle.

Reference Angles

A reference angle is the acute angle formed by the terminal side of an angle and the x-axis. It is used to simplify the calculation of trigonometric functions for angles greater than 90 degrees or less than 0 degrees. For example, the reference angle for 7π/6 is π/6, which helps in determining the sine value by considering the sign based on the quadrant in which the angle lies.

Trigonometric Function Values

Trigonometric function values, such as sine and cosine, can be derived from known angles on the unit circle. For instance, sin(7π/6) can be calculated using the reference angle π/6, which has a known sine value of 1/2. Since 7π/6 is in the third quadrant where sine values are negative, the exact value of sin(7π/6) is -1/2.