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

[3π/2, 2π] ; tan s = -1

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Identify the given interval for the variable \(s\), which is \(\left[\frac{3\pi}{2}, 2\pi\right]\), and the equation \(\tan s = -1\).

Recall that the tangent function has a period of \(\pi\), meaning \(\tan(s) = \tan(s + k\pi)\) for any integer \(k\). Also, tangent is negative in the second and fourth quadrants.

Determine the reference angle where \(\tan \theta = 1\). Since \(\tan s = -1\), the angle \(s\) must correspond to an angle where the tangent is \(1\) but with a negative sign.

Find the angles in the interval \(\left[\frac{3\pi}{2}, 2\pi\right]\) where \(\tan s = -1\). This interval corresponds to the fourth quadrant, where tangent is negative.

Express the solution(s) for \(s\) in terms of \(\pi\) using the reference angle \(\frac{\pi}{4}\), and verify that the solution(s) lie within the given interval.

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Unit Circle and Angle Intervals

The unit circle represents angles and their corresponding trigonometric values on a circle of radius 1. Understanding the interval [3π/2, 2π] means focusing on angles in the fourth quadrant, where the angle measures range from 270° to 360°, which affects the sign and value of trigonometric functions.

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Introduction to the Unit Circle

Tangent Function and Its Values

The tangent of an angle is the ratio of the sine to the cosine of that angle (tan θ = sin θ / cos θ). Knowing that tan s = -1 means the sine and cosine values have equal magnitude but opposite signs, which helps identify specific reference angles where this condition holds.

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Sine, Cosine, & Tangent of 30°, 45°, & 60°

Reference Angles and Sign of Trigonometric Functions

Reference angles are acute angles used to determine trigonometric values in different quadrants. Since tangent is negative in the fourth quadrant, the solution for s must correspond to an angle whose reference angle has tan = 1, but with a negative sign, guiding the exact value within the given interval.

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