Exploring the Trigonometric Functions When Cotangent is -1

Trigonometric functions, such as sine, cosine, tangent, secant, cosecant, and cotangent, are crucial in various fields, including physics, engineering, and mathematics. This article focuses on understanding these functions when the cotangent of an angle is -1. We will explore how this value influences the other five trigonometric functions.

Understanding the Cotangent Function

The cotangent function is defined as the ratio of the cosine to the sine of an angle:

(cot theta frac{cos theta}{sin theta})

If we are given that (cot theta -1), we can derive the values of the other five trigonometric functions. This condition implies that (cos theta -sin theta), suggesting that the angle (theta) is in either the second or the fourth quadrant.

Finding Sine and Cosine

To find the values of sine and cosine, we use the Pythagorean identity:

(sin^2 theta cos^2 theta 1)

Given that (cos theta -sin theta), we substitute:

(sin^2 theta (-sin theta)^2 1)

Thus, we get:

(2sin^2 theta 1)

Solving for (sin theta), we get:

(sin theta pm frac{sqrt{2}}{2})

Since (cos theta -sin theta), the corresponding cosine values are:

(cos theta -frac{sqrt{2}}{2}) in the second quadrant, and (cos theta frac{sqrt{2}}{2}) in the fourth quadrant.

Calculating the Other Trigonometric Functions

With the values of sine and cosine, we can calculate the other trigonometric functions as follows:

Sine

In the second quadrant:

(sin theta frac{sqrt{2}}{2})

In the fourth quadrant:

(sin theta -frac{sqrt{2}}{2})

Secant

In the second quadrant:

(sec theta frac{1}{cos theta} -sqrt{2})

In the fourth quadrant:

(sec theta frac{1}{cos theta} sqrt{2})

Tangent

The tangent function is given by the ratio of sine to cosine:

(tan theta frac{sin theta}{cos theta} -1)

Cosecant

The cosecant function is the reciprocal of sine:

In the second quadrant:

(csc theta frac{1}{sin theta} sqrt{2})

In the fourth quadrant:

(csc theta frac{1}{sin theta} -sqrt{2})

Cotangent

The cotangent is given as the reciprocal of tangent:

(cot theta -1)

Summary of Values

Based on the quadrant in which (theta) lies, we have the following values:

Second Quadrant:

(sin theta frac{sqrt{2}}{2}) (cos theta -frac{sqrt{2}}{2}) (tan theta -1) (sec theta -sqrt{2}) (csc theta sqrt{2}) (cot theta -1)

Fourth Quadrant:

(sin theta -frac{sqrt{2}}{2}) (cos theta frac{sqrt{2}}{2}) (tan theta -1) (sec theta sqrt{2}) (csc theta -sqrt{2}) (cot theta -1)

The appropriate set of values should be chosen based on the specific quadrant of (theta).