Tan(cos inverse 5x)
Is it set equal to anything?
Answer: (square root 1-25x^2 )/ 5x
To find the value of tan(cos^(-1)(5x)), we need to use some trigonometric identities and properties. Let's break it down step by step:
Step 1: Rewrite cos^(-1)(5x) using its definition
The notation cos^(-1)(5x) represents the arc cosine of 5x. This means we are looking for an angle whose cosine is equal to 5x. Let's call this angle θ.
So, cos^(-1)(5x) = θ
Step 2: Apply the definition of the tangent function
The tangent function can be defined as the ratio of the sine and cosine of an angle.
tan(θ) = sin(θ) / cos(θ)
Step 3: Determine the values for sin(θ) and cos(θ)
To find sin(θ) and cos(θ), we need to find the corresponding sides of a right triangle.
Since we know that cos(θ) = 5x, we can say that cos(θ) = adjacent side / hypotenuse.
Let's assume the adjacent side is represented by a and the hypotenuse by h.
Therefore, cos(θ) = a / h = 5x
Simplifying, we have a = 5x and h = 1 (since the hypotenuse is always 1 in the unit circle).
Using the Pythagorean theorem, we can find the opposite side of the triangle:
h^2 = a^2 + o^2
1^2 = (5x)^2 + o^2
1 = 25x^2 + o^2
o^2 = 1 - 25x^2
The opposite side is given by o = √(1 - 25x^2)
Step 4: Substitute the values into the tangent function
Now that we have the values for sin(θ) = o and cos(θ) = a, we can substitute them into the tangent formula: tan(θ) = sin(θ) / cos(θ).
tan(θ) = √(1 - 25x^2) / 5x
Therefore, tan(cos^(-1)(5x)) = √(1 - 25x^2) / 5x.