Question

Find the values of the six trigonometric functions for angle θ:

A. sin(θ), cos(θ), tan(θ), csc(θ), sec(θ), cot(θ)
B. cos(θ), sin(θ), cot(θ), sec(θ), csc(θ), tan(θ)
C. tan(θ), cot(θ), sin(θ), cos(θ), sec(θ), csc(θ)
D. csc(θ), sec(θ), tan(θ), sin(θ), cos(θ), cot(θ)

Answer 1

The six trigonometric functions—sin, cos, tan, csc, sec, cot—are ratios derived from the lengths of the sides of a right triangle relative to its angles. The sine and cosine functions are direct ratios, while the others are reciprocals or ratios of other functions.

Step-by-step explanation:

To find the values of the six trigonometric functions for an angle θ, we'll consider the right triangle definitions of these functions.

In a right triangle with angle θ, the following are the definitions of the six trigonometric functions:

σ(θ) = o/h — The sine of θ is the ratio of the length of the opposite side to the hypotenuse.

cos(θ) = a/h — The cosine of θ is the ratio of the length of the adjacent side to the hypotenuse.

tan(θ) = o/a — The tangent of θ is the ratio of the length of the opposite side to the adjacent side.

csc(θ) = 1/sin(θ) = h/o — The cosecant of θ is the reciprocal of the sine.

sec(θ) = 1/cos(θ) = h/a — The secant of θ is the reciprocal of the cosine.

cot(θ) = 1/tan(θ) = a/o — The cotangent of θ is the reciprocal of the tangent.

These functions are part of basic trigonometry, which deals with the relationships between angles and sides of triangles, more specifically right angled triangles.