Step by step solution to "Find the extrema of (1+sin⁡x)(1+cos⁡x) without using calculus."

High School
Geometry
Trigonometry
Trigonometric equations and identities

Shirley Thompson

Answered question

2022-01-02

Find the extrema of

(1 + \sin x) (1 + \cos x)

without using calculus.

Answer & Explanation

Anzante2m

Beginner2022-01-03Added 34 answers

Hint:

(1 + \sin x) (1 + \cos x) = 1 + \sin x + \cos x + \frac{12}{\sin 2} x = 1 + \sqrt{2} \sin (x + \frac{π}{4}) + \frac{12}{\sin 2} x

Jenny Sheppard

Beginner2022-01-04Added 35 answers

$(1 + \sin x) (1 + \cos x) = 1 + \sin x + \cos x + \sin x \cos x$
Note that $\sin x \cos x = \frac{1}{2} \sin 2 x$
Note also that
$\sin x + \cos x = \sqrt{2} (\frac{\sqrt{2}}{2} \sin x + \frac{\sqrt{2}}{2} \cos x) = \sqrt{2} (\cos \frac{π}{4} \sin x + \sin \frac{π}{4} \cos x) = \sqrt{2} \sin (x + \frac{π}{4})$
Both of these factors reach their maximum when $x = \frac{π}{4}$ , so the maximum is $\frac{32}{+} \sqrt{2}$

Vasquez

Expert2022-01-08Added 669 answers

$\begin{matrix} N o t e \\ A = (1 + \sin x) (1 + \cos x) \\ = 1 + \sin x + \cos x + \frac{1}{2} \sin 2 x \\ = 1 + \sqrt{2} \cos (\frac{π}{4} - x) + \frac{1}{2} \cos (\frac{π}{2} - 2 x) \\ = \frac{1}{2} + \sqrt{2} \cos (\frac{π}{4} - x) + \cos^{2} (\frac{π}{4} - x) \\ = (\cos (\frac{π}{4} - x) + \frac{1}{\sqrt{2}})^{2} \\ T h u s \\ 0 \leq A \leq (1 + \frac{1}{\sqrt{2}})^{2} \end{matrix}$

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