Question

Two rectangular parks R1,R2 are decorated with fencing. If the length of the fencing of R2 is time of the length fencing of R1, find
(a)Formula for perimeter of rectangle
(b) Find the perimeter of R2 if the length of R1 fencing is 20cm

Answer 1

Answer:

EXPLANATION.

Extreme value of,$\sf \implies 4cos(x^{2} ). cos\bigg(\dfrac{\pi}{3} + x^{2}\bigg).cos\bigg(\dfrac{\pi}{3} - x^{2} \bigg)$

As we know that,Formula of : cos(A ± B).

⇒ Cos(A + B) = Cos(A).Cos(B) - Sin(A).Sin(B).

⇒ Cos(A - B) = Cos(A).Cos(B) + Sin(A).Sin(B).

Using the formula in equation, we get.

$\sf \implies 4cos( x^{2} ) \bigg(cos\dfrac{\pi}{3} .cos(x^{2} ) - sin\dfrac{\pi}{3} .sin(x^{2} ) \bigg). \bigg(cos\dfrac{\pi}{3}.cos(x^{2} ) + sin\dfrac{\pi}{3}.sin(x^{2} ) \bigg)$

As we know that,

Formula of :⇒ cos(π/3) = cos(180/3) = cos(60°) = 1/2.

⇒ sin(π/3) = sin(180/3) = sin(60°) = √3/2.

Using the formula in equation, we get.

$\sf \implies 4cos(x^{2} ) \bigg(\dfrac{1}{2}cos(x^{2} ) -\dfrac{\sqrt{3} }{2} sin(x^{2} ) \bigg). \bigg(\dfrac{1}{2}cos(x^{2} ) + \dfrac{\sqrt{3} }{2}sin(x^{2} ) \bigg)$

As we know that,

Formula of :⇒ (a² - b²) = (a + b)(a - b).

Using the formula in equation, we get.

$\sf \implies 4cos(x^{2} ) \bigg(\dfrac{1}{2}cos(x^{2} ) \bigg)^{2} .- \bigg(\dfrac{\sqrt{3} }{2} sin(x^{2}) \bigg)^{2}$

$\sf \implies 4cos(x^{2} ) \bigg(\dfrac{1}{4} cos^{2}x^{2} - \dfrac{3}{4}sin^{2} x^{2} \bigg)$

$\sf \implies \dfrac{4cos(x^{2} )}{4} \bigg(cos^{2} x^{2} - 3sin^{2} x^{2} \bigg)$

As we know that,

Formula of :

⇒ sin²∅ + cos²∅ = 1.⇒ sin²∅ = 1 - cos²∅.

Using this formula in equation, we get.

$\sf \implies cos(x^{2} ) \bigg(cos^{2}x^{2} - 3(1 - cos^{2} x^{2} ) \bigg)$

$\sf \implies cos^{} (x^{2} )\bigg(cos^{2} x^{2} -3+3cos^{2} x^{2} \bigg)$

$\sf \implies cos(x^{2} ) \bigg(4cos^{2}x^{2} - 3\bigg)$

$\sf \implies 4cos^{3} (x^{2}) - 3cos(x^{2})$

As we know that,

Formula of :

⇒ cos3∅ = 4cos³∅ - 3cos∅.

Using this formula in equation, we get.

⇒ cos(3x²).

As we know that,

Range of cos∅.⇒ cos∅ = -1 < cos∅ < 1.

⇒ range = [-1,1].

So,⇒ cos3x² = -1 < cos3x² < 1.

Extreme value of cos(3x²) = [ -1, 1].

Option [A] is correct answer.

                                                                                                                                      MORE INFORMATION.

Domain & Range of inverse trigonometric functions.

(1) = sin⁻¹x

Domain = [-1, 1].

Range = [ -π/2, π/2].

(2) = cos⁻¹x

Domain = [-1, 1].

Range = [0, π].

(3) = tan⁻¹x

Domain = (-∞. ∞).

Range = (-π/2, π/2).

(4) = cot⁻¹x

Domain = (-∞, ∞).

Range = (-π/2, π/2).

(5) = sec⁻¹x

Domain = (-∞, -1] ∪ [1,∞).

Range = [0,π/2) ∪ (π/2, π].

(6) = cosec⁻¹x

Domain = (-∞, -1] ∪ [1,∞).

Range = [-π/2, 0 ) ∪ (0, π/2].

Answer 2

Answer:

Two rectangular parks R1,R2 are decorated with fencing. If the length of the fencing of R2 is time of the length fencing of R1, find

(a)Formula for perimeter of rectangle

(b) Find the perimeter of R2 if the length of R1 fencing is 20cm

Step-by-step explanation:

Two rectangular parks R1,R2 are decorated with fencing. If the length of the fencing of R2 is time of the length fencing of R1, find

(a)Formula for perimeter of rectangle

(b) Find the perimeter of R2 if the length of R1 fencing is 20cm