Find each exact function value. See Example 2.
cos 7π/4

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the exact value of \( \cos \left( \frac{7\pi}{4} \right) \).
2. Step 2: Recognize that \( \frac{7\pi}{4} \) is an angle in radians. Convert it to degrees if necessary for better understanding. \( \frac{7\pi}{4} \) radians is equivalent to 315 degrees.
3. Step 3: Determine the quadrant in which the angle lies. Since 315 degrees is between 270 and 360 degrees, it lies in the fourth quadrant.
4. Step 4: Recall the properties of cosine in the fourth quadrant. In the fourth quadrant, the cosine of an angle is positive.
5. Step 5: Use the reference angle to find the cosine value. The reference angle for 315 degrees is 360 - 315 = 45 degrees. Therefore, \( \cos \left( \frac{7\pi}{4} \right) = \cos(45^\circ) = \frac{\sqrt{2}}{2} \).

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 x-coordinate corresponds to cos(θ) and the y-coordinate corresponds to sin(θ).

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/4 is π/4, which helps in determining the cosine value.

Trigonometric Values in Quadrants

Trigonometric functions have different signs depending on the quadrant in which the angle lies. In the unit circle, angles in the fourth quadrant (like 7π/4) have a positive cosine value and a negative sine value. Understanding which quadrant an angle is in is crucial for determining the correct sign of the trigonometric function values.