Rewrite: tan(cos^−1 (v)) as an algebraic expression in v.
Please help me understand.
cos theta = adjacent/hypotenuse
call adjacent side v
call hypotenuse 1
then cos theta = v
opposite side = sqrt(1^2 - v^2)
tan theta = sqrt(1-v^2) /v
To rewrite tan(cos^−1 (v)), we need to express it as an algebraic expression in terms of v.
1. Let's start by understanding the given expression cos^−1 (v). The notation cos^−1 (v) represents the inverse cosine function, also known as arccosine or acos. The acos function gives us the angle whose cosine is v. So, cos^−1 (v) = acos(v).
2. Next, we substitute this value of cos^−1 (v) in the original expression: tan(cos^−1 (v)) = tan(acos(v)).
3. The trigonometric function tan is defined as the ratio of sine to cosine: tan(x) = sin(x) / cos(x).
4. To express tan(acos(v)) in terms of v, we need to find the corresponding values of sin(x) and cos(x) for acos(v).
5. We can use the trigonometric identity sin^2(x) + cos^2(x) = 1 to find the value of sin(x) using the value of cos(x).
6. Since we know that cos(x) = v (from acos(v)), we can substitute cos(x) with v in the identity: sin^2(x) + v^2 = 1.
7. Solving this equation for sin(x), we get: sin(x) = sqrt(1 - v^2).
8. Substituting sin(x) and cos(x) in tan(acos(v)) with their respective values, we have: tan(acos(v)) = (sqrt(1 - v^2)) / v.
Therefore, the algebraic expression for tan(cos^−1 (v)) in terms of v is: (sqrt(1 - v^2)) / v.
To rewrite the expression tan(cos^−1(v)) as an algebraic expression in v, we need to use trigonometric identities and properties.
First, we recall the identity: cos^−1(v) = α, where α is the angle whose cosine is v. This means that v = cos(α).
Now, we can rewrite the expression tan(cos^−1(v)) using the angle α:
tan(α)
Next, we need to find an expansion for tan(α) in terms of v. To do this, we can use the identity: tan^2(α) = 1 - cos^2(α). This identity can be rewritten as: tan^2(α) + cos^2(α) = 1.
Since v = cos(α), we can substitute this into the identity:
tan^2(α) + v^2 = 1
From here, we solve for tan(α):
tan(α) = ±√(1 - v^2)
So, rewriting tan(cos^−1(v)) as an algebraic expression in v is:
±√(1 - v^2)
Note that the ± sign accounts for multiple possible solutions, as both positive and negative values of √(1 - v^2) are valid depending on the quadrant in which α lies.