What are the six trigonometric function values of 540?

To determine the six trigonometric function values for 540 degrees, let's break it down step by step.

Convert 540° into a standard angle:

540° is more than 360°, so we need to subtract 360° to find an equivalent angle within the range of 0° to 360°.
540° - 360° = 180°.
Thus, the reference angle is 180°.

Determine the trigonometric functions: The angle 180° lies on the negative x-axis in the unit circle. Let's evaluate each of the six trigonometric functions at 180°.

a. Sine (sin 180°):

The sine of an angle is the y-coordinate of the corresponding point on the unit circle.
At 180°, the point on the unit circle is (-1, 0).
Therefore, sin(180°) = 0.
b. Cosine (cos 180°):

The cosine of an angle is the x-coordinate of the corresponding point on the unit circle.
At 180°, the point on the unit circle is (-1, 0).
Therefore, cos(180°) = -1.
c. Tangent (tan 180°):

The tangent of an angle is the ratio of sine to cosine (tan(θ) = sin(θ) / cos(θ)).
At 180°, sin(180°) = 0 and cos(180°) = -1.
Therefore, tan(180°) = 0 / -1 = 0.
d. Cosecant (csc 180°):

The cosecant is the reciprocal of sine (csc(θ) = 1 / sin(θ)).
At 180°, sin(180°) = 0, so csc(180°) is undefined (since division by 0 is undefined).
e. Secant (sec 180°):

The secant is the reciprocal of cosine (sec(θ) = 1 / cos(θ)).
At 180°, cos(180°) = -1.
Therefore, sec(180°) = 1 / -1 = -1.
f. Cotangent (cot 180°):

The cotangent is the reciprocal of tangent (cot(θ) = 1 / tan(θ)).
At 180°, tan(180°) = 0, so cot(180°) is undefined (since division by 0 is undefined).
Summary of the six trigonometric function values for 540°:

sin(540°) = 0
cos(540°) = -1
tan(540°) = 0
csc(540°) = undefined
sec(540°) = -1
cot(540°) = undefined