Write as an algebraic expression in u.
Tan(cos^-1 u)
right triangle
adjacent = u
hypotenuse = 1
so opposite = sqrt(1-u^2
so tan = opposite/adjacent = sqrt(1-u^2)/u
Oh okay thanks!
Cos²∅= 13÷12.find 1+cot²∅
To write the expression "Tan(cos^-1 u)" in terms of u, we can first start by understanding the meaning and relationship between the trigonometric functions involved.
The inverse cosine (cos^-1) function is also known as arccos or cos^-1. It takes an input value and returns the angle whose cosine equals that input value. The output of cos^-1 u represents an angle, which we can represent as θ.
The tangent (Tan) function is defined as the ratio of the sine of an angle to the cosine of the same angle. Mathematically, Tan θ = sin θ / cos θ.
Now let's proceed to solve the expression "Tan(cos^-1 u)" algebraically.
Step 1: Let's assume cos^-1 u = θ.
Step 2: We know that cos θ = u. This is the equivalent of cos^-1 u = θ.
Step 3: To find the value of sin θ, we can use the Pythagorean Identity: sin^2 θ + cos^2 θ = 1.
Step 4: Since cos^2 θ = u^2, substituting it into the Pythagorean Identity gives: sin^2 θ + u^2 = 1.
Step 5: Solving for sin θ, we get: sin θ = sqrt(1 - u^2).
Step 6: Now we can substitute sin θ and cos θ into the tangent equation: Tan θ = sin θ / cos θ.
Step 7: Replacing sin θ with sqrt(1 - u^2) and cos θ with u, the expression becomes:
Tan(cos^-1 u) = (sqrt(1 - u^2)) / u.
Therefore, the algebraic expression for Tan(cos^-1 u) is (sqrt(1 - u^2)) / u.