Find the exact value of s in the given interval that has the given circular function value.


[3π/2, 2π] ; tan s = -1

Verified step by step guidance

1. Identify the interval [\frac{3\pi}{2}, 2\pi] and note that it corresponds to the fourth quadrant of the unit circle.
2. Recall that the tangent function, \( \tan s = \frac{\sin s}{\cos s} \), is negative in the fourth quadrant.
3. Recognize that \( \tan s = -1 \) implies that the sine and cosine values are equal in magnitude but opposite in sign.
4. Determine the reference angle where \( \tan \theta = 1 \), which is \( \theta = \frac{\pi}{4} \).
5. Find the angle \( s \) in the fourth quadrant by subtracting the reference angle from \( 2\pi \), resulting in \( s = 2\pi - \frac{\pi}{4} \).

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 provides a geometric representation of the sine, cosine, and tangent functions. The angles on the unit circle correspond to points on the circle, allowing for the determination of trigonometric values based on the coordinates of these points.

Tangent Function

The tangent function, defined as the ratio of the sine to the cosine (tan θ = sin θ / cos θ), represents the slope of the line formed by the angle θ in the unit circle. It is periodic with a period of π, meaning it repeats its values every π radians. Understanding the behavior of the tangent function is crucial for solving equations involving it, especially when determining angles in specific intervals.

Quadrants of the Unit Circle

The unit circle is divided into four quadrants, each corresponding to different signs of the sine, cosine, and tangent functions. In the third quadrant (from π to 3π/2), both sine and cosine are negative, making tangent positive. In the fourth quadrant (from 3π/2 to 2π), sine is negative and cosine is positive, resulting in a negative tangent. Recognizing the quadrant in which an angle lies is essential for determining the correct angle that satisfies a given trigonometric equation.