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2 years ago

Need help on the Sin(4x) part of the question please!!

Mathematics

1 answer:

Ksivusya [100]2 years ago

6 0

a. The equivalent expression for cos(4x) in terms of just x is 8cos⁴x - 8cos²x + 1

b. The equivalent expression for sin(4x) in terms of just x is 4sinxcos³x - 2sinxcosx

<h3>a. The equivalent trigonometric expression for cos(4x)</h3>

To find the trigonometric expression for cos(4x)

cos(4x) = cos[2(2x)]

Now from trigonometric identities

cos2Ф = 2cos²Ф - 1

Let 2x = Ф.

So, cos(4x) = 2cos²2x - 1

Also, cos2x = 2cos²x - 1

So, substituting cos2x into cos4x, we have

cos(4x) = 2cos²2x - 1

cos(4x) = 2[2cos²x - 1]² - 1

cos(4x) = 2[4cos⁴x - 4cos²x + 1] - 1

cos(4x) = 8cos⁴x - 8cos²x + 2 - 1

cos(4x) = 8cos⁴x - 8cos²x + 1

So, the equivalent expression for cos(4x) in terms of just x is 8cos⁴x - 8cos²x + 1

<h3>b. The equivalent trigonometric expression for sin(4x)</h3>

To find the expression for sin(4x)

sin(4x) = sin2(2x)

From trigonometric identities sin2Ф = 2sinФcosФ

Let 2x = Ф

So, sin(4x) = sin2(2x)

= 2sin2xcos2x

Since

sin2x = 2sinxcosx and
cos2x = 2cos²x - 1

Substituting these into the equation, we have

sin(4x) = 2sin2xcos2x

sin(4x) = (2sinxcosx)(2cos²x - 1)

sin(4x) = 4sinxcos³x - 2sinxcosx

So, the equivalent expression for sin(4x) in terms of just x is 4sinxcos³x - 2sinxcosx

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The revenue function is a quadratic equation and the graph of the function

has the shape of a parabola that is concave downwards.

The correct responses are;

(a) R = -x² + 82·x

(b) $1,645

(c) The graph of R has a maximum because the leading coefficient of the quadratic function for R is negative.

(d) R = -1·(x - 41)² + 1,681

(e) 41

(f) $1,681

Reasons:

The given function that gives the weekly revenue is; R = x·(82 - x)

Where;

R = The revenue in dollars

x = The number of lunches

(a) The revenue can be written in the form R = a·x² + b·x + c by expansion of the given function as follows;

R = x·(82 - x) = 82·x - x²

Which gives;

R = -x² + 82·x

Where, the constant term, c = 0

(b) When 35 launches are sold, we have;

x = 35

Which by plugging in the value of x = 35, gives;

R = 35 × (82 - 35) = 1,645

The revenue when 35 lunches are sold, R = $1,645

Given that the leading coefficient is negative, the shape of graph of the

function R is concave downward, and therefore, the graph has only a

maximum point.

(d) The form a·(x - h)² + k is the vertex form of quadratic equation, where;

(h, k) = The vertex of the equation

a = The leading coefficient

The function, R = x·(82 - x), can be expressed in the form a·(x - h)² + k, as follows;

R = x·(82 - x) = -x² + 82·x

At the vertex, of the equation; f(x) = a·x² + b·x + c, we have;

$\displaystyle x = \mathbf{-\frac{b}{2 \cdot a}}$

Therefore, for the revenue function, the x-value of the vertex, is; $\displaystyle x = -\frac{82}{2 \times (-1)} = \mathbf{41}$

The revenue at the vertex is; $R_{max}$ = 41×(82 - 41) = 1,681

Which gives;

(h, k) = (41, 1,681)

a = -1 (The coefficient of x² in -x² + 82·x)

The revenue equation in the form, a·(x - h)² + k is; R = -1·(x - 41)² + 1,681

(e) The number of lunches that must be sold to achieve the maximum revenue is given by the x-value at the vertex, which is; x = 41

Therefore;

The number of lunches that must be sold for the maximum revenue to be achieved is 41 lunches

(f) The maximum revenue is given by the revenue at the vertex point where x = 41, which is; R = $1,681

The maximum revenue of the company is $1,681

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