How do you find the exact values of cot, cosec and sec for 180 degrees?

To find the exact values of cotangent (cot), cosecant (cosec), and secant (sec) for 180 degrees, we can use the definitions of these trigonometric functions based on the unit circle.

Understanding the Angle

180 degrees corresponds to a point on the unit circle where the coordinates are (-1, 0). This is important for calculating the trigonometric functions.

Calculating Cotangent

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

• cot(θ) = cos(θ) / sin(θ)

For 180 degrees:

• cos(180°) = -1
• sin(180°) = 0

Since division by zero is undefined, cot(180°) is undefined.

Finding Cosecant

Cosecant is the reciprocal of sine:

• cosec(θ) = 1 / sin(θ)

For 180 degrees:

• sin(180°) = 0

Again, division by zero means cosec(180°) is undefined.

Determining Secant

Secant is the reciprocal of cosine:

• sec(θ) = 1 / cos(θ)

For 180 degrees:

• cos(180°) = -1

Thus, we find:

• sec(180°) = 1 / -1 = -1

Summary of Values

In summary, the values for cotangent, cosecant, and secant at 180 degrees are:

• cot(180°) = undefined
• cosec(180°) = undefined
• sec(180°) = -1