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Leona [35]
3 years ago
9

Let t = cosA + sinA find values of t if 2 - sin2A = 2(cosA + sinA)

Mathematics
1 answer:
Zinaida [17]3 years ago
4 0

2 - \sin(2A) = 2(\cos A + \sin A)\\\\2-2\sin A\cos A=2\cos A+2\sin A\\\\2\sin A\cos A+2\cos A+2\sin A-2=0\\\\1+2\sin A\cos A+2\cos A+2\sin A-3=0\\\\\sin^2 A+\cos^2 A+2\sin A\cos A+2\cos A+2\sin A-3=0\\\\(\sin A+\cos A)^2+2\cos A+2\sin A-3=0\\\\(\sin A+\cos A)^2+2(\cos A+\sin A)-3=0\\\\\\t^2+2t-3=0\\\\t^2+2t+1-4=0\\\\(t+1)^2=4\\\\t+1=2\qquad\vee \qquad t+1=-2\\\\\\\boxed{t=1\qquad\vee\qquad t=-3}

When A\in\mathbb{R}

t=\cos A+\sin A\in[-\sqrt{2};\sqrt{2}]\approx[-1.4142;1.4142]

and there is only one answer t = 1.

For A\in\mathbb{C} both values are correct.

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