let x>0. write the expression as an algebraic expression in x: csc(arctan(x))
To write the expression csc(arctan(x)) as an algebraic expression in x, follow these steps:
Step 1: Recall the definitions of csc and arctan functions:
- csc(theta) = 1/sin(theta)
- arctan(x) is the angle whose tangent is x
Step 2: Substitute arctan(x) with its definition:
csc(arctan(x)) = 1/sin(arctan(x))
Step 3: Determine the value of sin(arctan(x)):
- Let's draw a right triangle to help visualize this.
- Assume that the angle whose tangent is x is theta. Therefore, tan(theta) = x.
- Let's label the opposite side of theta as x and the adjacent side as 1 (since tan(theta) = x).
- Use the Pythagorean theorem to find the hypotenuse: hypotenuse^2 = opposite^2 + adjacent^2
=> hypotenuse^2 = x^2 + 1^2
=> hypotenuse = sqrt(x^2 + 1)
Step 4: Determine the value of sin(theta):
- sin(theta) = opposite/hypotenuse
- sin(theta) = x/sqrt(x^2 + 1)
Step 5: Substitute sin(arctan(x)) with its value:
csc(arctan(x)) = 1/sin(arctan(x))
= 1 / [x/sqrt(x^2 + 1)]
= sqrt(x^2 + 1)/x
Therefore, the algebraic expression csc(arctan(x)) can be written as sqrt(x^2 + 1)/x.
To write the expression csc(arctan(x)) as an algebraic expression, we need to simplify it using trigonometric identities.
In a right triangle, the cosecant (csc) of an angle is defined as the reciprocal of the sine of that angle. Similarly, the inverse tangent (arctan) of a value returns the angle whose tangent is equal to that value.
First, let's consider the triangle where one of the acute angles is denoted as θ, and let's assume that the opposite side is x, and the adjacent side is 1.
Using the definition of trigonometric functions, we have:
sin(θ) = x/hypotenuse = x/√(1+x^2) (using Pythagorean theorem)
csc(θ) = 1/sin(θ) = √(1+x^2)/x
Therefore, csc(arctan(x)) can be written as √(1+x^2)/x, where x > 0.
Draw right triangle ABC, with sides labeled so that tanA = x/1
you can see that cscA = √(x^2+1)/x
If arctan ( x ) = θ
then
tan θ = x
cot θ = 1 / tan θ = 1 / x
csc² θ = 1 + cot² θ = 1+ ( 1 / x )² = 1 + 1 / x² =
x² / x² + 1 / x² = ( x² + 1 ) / x²
csc θ = √( x² + 1 ) / √x²
csc θ = √( x² + 1 ) / x
So
csc ( arctan ( x ) ) = √( x² + 1 ) / x