Question

Find the exact values of s in the interval [-2π ,π ) that satisfy the given condition cot² s=1.

Answer 1

The exact values of s in the interval [-2π, π) that satisfy cot² s = 1 are π/4, -3π/4, and -7π/4.

Step-by-step explanation:

We are looking to find the exact values of s in the interval [-2π, π) for which cot² s = 1. To solve this, we will look for angles where the cotangent of the angle is ±1 since squaring ±1 gives us 1.

Cotangent is the reciprocal of tangent, so where tangent is ±1, cotangent will also be ±1. The solutions for tangent of angle s to be ±1 are angles where sine and cosine have the same absolute value but may differ in sign (since tangent is sine divided by cosine). This occurs at angle s = π/4 and s = -3π/4 (or equivalent rotations).

If we look at the given interval, the following angles satisfy cot² s = 1: -3π/4, π/4, and their coterminal angles within the interval. However, since the interval does not include π but does include -2π, -7π/4 is an additional solution which is coterminal with π/4.

The exact values of s within the given interval [-2π, π) that satisfy cot² s = 1 are π/4, -3π/4, and -7π/4.

.

Answer 2

To find the values of s that satisfy the equation cot² s = 1 in the interval [-2π, π), manipulate the equation to find tan s = 1 and determine the values of s that satisfy the equation.

Step-by-step explanation:

To find the exact values of s in the interval [-2π, π) that satisfy the condition cot² s = 1, we first need to manipulate the equation to obtain the value of cot s and then find the values of s that satisfy the equation.

Since cot s = 1, we know that tan s = 1/cot s = 1. In the given interval, tan s is positive in the second and fourth quadrants where sine is positive. Therefore, the values of s that satisfy the equation are π/4 and 5π/4.