find the least positive value of @. sec (4@+10)cos(3@+17)
@=theta
To find the least positive value of @ that minimizes the expression sec(4@+10)cos(3@+17), we can follow these steps:
Step 1: We need to find the critical points for the expression. In this case, the critical points occur when the derivative of the expression is equal to zero.
Step 2: Calculate the derivative of the given expression with respect to @. The derivative of sec(4@+10)cos(3@+17) can be found using the chain rule and product rule.
Let's go through the steps to find the derivative:
Step 2a: Derivative of sec(4@+10)
Since sec(θ) is equal to 1/cos(θ), the derivative of sec(4@+10) is -(4sec(4@+10)tan(4@+10)).
Step 2b: Derivative of cos(3@+17)
The derivative of cos(3@+17) is -3sin(3@+17).
Step 2c: Product rule
To find the derivative of a product of functions, we apply the product rule:
(d/d@)[sec(4@+10)cos(3@+17)] = [cos(3@+17)] * [-(4sec(4@+10)tan(4@+10))] + [sec(4@+10)] * [-(3sin(3@+17))]
Step 3: Set the derivative equal to zero and solve for @.
Set the derivative equal to zero:
[cos(3@+17)] * [-(4sec(4@+10)tan(4@+10))] + [sec(4@+10)] * [-(3sin(3@+17))] = 0
Now, we need to solve this equation to find the critical points. This equation may not have a simple algebraic solution, so we will need to use numerical methods or a graphing calculator to find the approximate values of @ that satisfy the equation.
Step 4: Once you have obtained the critical points, substitute these points back into the original expression and determine the minimum value of the expression.
Let's calculate the approximate critical points and find the minimum value of the expression using a graphing calculator or numerical methods.