what is the cofunction and reciprocal of the cosecant?
there are 3 pairs of cofunctions
sine vs cosine
tangent vs cotangent
secant vs cosecant
Surely your text or notes have definitions of the reciprocal functions.
so would it be y=sin x for the reciprocal? and y=sec x for cofunction?
The cosecant function (csc) is defined as the reciprocal of the sine function (sin).
Therefore, the cofunction of the cosecant is the sine function:
Cofunction of csc(x) = sin(x)
And the reciprocal of the cosecant is obtained by taking the reciprocal of the sine function:
Reciprocal of csc(x) = 1/sin(x)
To understand the cofunction and reciprocal of the cosecant, let's start with the cosecant itself.
The cosecant function, denoted as csc(x), is the reciprocal of the sine function (sin(x)). It is defined as the ratio of the hypotenuse to the length of the side opposite a given angle in a right triangle.
Now, let's address the cofunction. In trigonometry, the cofunction of an angle is the function of the complement of that angle. The complement of an angle is the difference between that angle and a right angle (90 degrees or π/2 radians).
So, to find the cofunction of the cosecant:
1. Start with the cosecant function, csc(x).
2. Determine the complement of the angle, which is 90 degrees minus the given angle, or π/2 - x.
3. Identify the reciprocal of the function for the complement angle. Since csc(x) is the reciprocal of sin(x), the cofunction of csc(x) will be the reciprocal of sin(π/2 - x).
Therefore, the cofunction of the cosecant is the secant function (sec(x)). It is the reciprocal of the cosine function (cos(x)).
To summarize:
- The cofunction of the cosecant is the secant (sec(x)).
- The reciprocal of the cosecant is the same as the reciprocal of the sine, so it is equal to the cosecant itself (csc(x)).