How to find the exact value of
tan(sin^-1x/2)
I'm having trouble with finding the inverse of sin x/2 value.
To find the exact value of tan(sin^-1(x/2)), we'll start by understanding how to find the inverse of sin(x/2) and then evaluate the tangent of that inverse value.
1. Finding the inverse of sin(x/2):
To find the inverse of sin(x/2), we can rewrite it as sin^-1(x/2) or arcsin(x/2). The inverse of a trigonometric function "undoes" the original function. In this case, it gives us the angle whose sine is x/2.
2. Evaluating the inverse trigonometric function:
To evaluate the inverse trigonometric function, we need to know the range of values it can take. The sine function oscillates between -1 and 1, so the domain of sin^-1 is limited to the interval [-1, 1]. The output of sin^-1(x/2) will also fall within this interval.
Now, to find the inverse sin^-1(x/2), you need to ask yourself: "What angle, when plugged into the sine function, gives me x/2?"
Since we're dealing with sin(x/2), we can rewrite it as x/2 = sin(θ), where θ represents the angle whose sine is x/2.
To solve for θ, take the inverse sine (or arcsine) of both sides:
sin^-1(x/2) = θ
So, the value of θ is the inverse sine of x/2.
3. Evaluating tangent of the inverse value:
Now that we have the value of θ, let's find the value of tan(sin^-1(x/2)).
The tangent of an angle is defined as the ratio of the sine of the angle to the cosine of the angle. In this case, we have:
tan(sin^-1(x/2)) = sin(sin^-1(x/2)) / cos(sin^-1(x/2))
Using the identity sin^2θ + cos^2θ = 1, we can rewrite the above expression:
tan(sin^-1(x/2)) = x/2 / cos(sin^-1(x/2))
To evaluate this expression, we need to know the value of cos(sin^-1(x/2)). However, finding the exact value of cos(sin^-1(x/2)) may require more information.
If you have additional constraints or specific values for x, we can provide a more accurate evaluation.