Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

$https://latex.codecogs.com/png.image?\inline \large \dpi{120}\\\text{Als}\\ {\color{Green} .}\qquad \frac{\sin 2\alpha }{\sin\alpha }=-\frac 12\\\text{dan is}\\{\color{Green} .}\qquad \frac{\cos 2\alpha }{\cos\alpha }\\\text{gelijk aan}$	A. − 1
	B. 2
	C. 3,5
	D. 4,5
	E. − 0,5

[ 5-4018 - op net sinds 12.10.12-(E)-29.11.2023 ]

Translation in E N G L I S H

$\\\text{ If}\\ {\color{White} .}\qquad \frac{\sin 2\alpha }{\sin\alpha }=-\frac 12\\\text{then}\\{\color{White} .}\qquad \frac{\cos 2\alpha }{\cos\alpha }\\\text{equals}$	A. − 1
	B. 2
	C. 3.5
	D. 4.5
	E. − 0.5

Oplossing - Solution

$\\\frac{\sin2\alpha}{\sin\alpha}=-\frac12\;\;\Leftrightarrow\;\;\frac{2.\sin\alpha.\cos\alpha}{\sin\alpha}=-\frac12\;\;\Leftrightarrow\;\;2\cos\alpha=-\frac12\;\;\Leftrightarrow\;\;\cos\alpha=-\frac14\\1+\cos2\alpha=2\cos^2\alpha\;\;\Leftrightarrow\;\cos2\alpha=2\cos^2\alpha-1=2\left(-\frac14\right)^2-1=\frac18-1=\frac78\\\therefore\frac{\cos2\alpha}{\cos\alpha}=\frac{-\frac78}{-\frac14}=\frac78.4=\frac72$