Write an algebraic expression for y=tan (cos^-1 x/3)for 0<x<3.
Please Help!!
look at the cos^-1 (x/3)
construct a triangle with base of x and hypotenuse of 3
by Pythagoras:
r^2 = x^2 + y^2
3^2 = x^2 + y^2
y^2 = 9 - x^2
y = √(9-x^2)
so we want the tangent
tan (cos^-1 x/3)
= opposite/adjacent
= √(9-x^2)/3
To write the algebraic expression for y = tan(cos^(-1)(x/3)), we'll break it down step-by-step:
Step 1: Start with the innermost operation, cos^(-1)(x/3), which represents the inverse cosine of (x/3).
Step 2: Since the domain of arccosine is between -1 and 1, we need to ensure that x/3 is within this range. Given the condition 0 < x < 3, we can deduce that 0/3 < x/3 < 3/3 (0 < x/3 < 1). Therefore, we can consider arccosine as a valid operation for the given domain.
Step 3: Next, we find the tangent of the value obtained in Step 2, which is the result of the inverse cosine operation.
Putting it all together, the algebraic expression becomes:
y = tan(cos^(-1)(x/3)) for 0 < x < 3
To write the algebraic expression for y = tan(cos^(-1)(x/3)), we need to break down the expression step by step.
Step 1: Find the value of cos^(-1)(x/3):
The inverse cosine function (cos^(-1)) returns the angle whose cosine is x/3. Since 0 < x < 3, x/3 will also be within the range of -1 to 1, which is the acceptable range for the cosine function. We can denote cos^(-1)(x/3) as angle θ.
Step 2: Find the value of tan(θ):
Once we have the value of θ, we can substitute it into the tangent function. The tangent function (tan) returns the ratio of the sine and cosine of an angle. So, we need to convert θ back into the sine and cosine values.
Step 3: Solve for y:
Since θ is the result of cos^(-1)(x/3) and y is the result of tan(θ), we can substitute θ into the expression for y. The final expression will be in terms of x.
Let's go ahead and solve it step by step.
Step 1: Find the value of cos^(-1)(x/3).
To find the angle whose cosine is x/3, we can use the inverse cosine function. In other words, we want to find the value of θ such that cos(θ) = x/3. To calculate this value in radians, you can use the arccosine function (acos) on most scientific calculators or math software.
Step 2: Find the value of tan(θ).
Once we have θ, we can substitute it into the tangent function to find y. We can calculate this using the tangent function (tan) on a calculator or math software.
Step 3: Solve for y.
Finally, substitute the value of θ calculated in step 1 into the expression for y = tan(θ). Simplify the expression if necessary.
Keep in mind that the resulting expression for y might be in terms of x and θ.