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3 years ago

Lin multiplies 7/8 times a number.the product is less than 7/8 . Which could be Lin’s number ?

Mathematics

1 answer:

igor_vitrenko [27]3 years ago

3 0

Answer:

Any number less than 1

Step-by-step explanation:

Let x be the number Lin multiplies by 7/8. So, we have:

(7/8)*x < 7/8

We need to find the possible values for x. Lets start by dividing both sides of our inequality by (7/8) in order to get x alone on the left side.

(7/8)*x / (7/8) < (7/8)/(7/8)

Reordering:

(7/8) / (7/8) * x < (7/8)/(7/8)

As any number divided by itself (except from 0) is equal to 1:

x < 1

So, the number that Lin multiplied by 7/8 MUST be less than 1. This is evident, as any positive number multiplied by a number less than 1 will decrease, doesn't matter if its a negative or positive number.

So, such number is less than 1. Any number less than 1 could be Lin's number.

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Write -3+4i in the form r(cosx+isinx)

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$-3+4i$ is a complex number that satisfies

$\begin{cases}r\cos x=-3\\[1ex]r\sin x=4\\[1ex]r=\sqrt{(-3)^2+4^2}\end{cases}$

The last equation immediately tells you that $r=5$ .

So you have

$\begin{cases}\cos x=-\dfrac35\\[1ex]\sin x=\dfrac45\end{cases}$

Dividing the second equation by the first, you end up with

$\dfrac{\sin x}{\cos x}=\tan x=\dfrac{\dfrac45}{-\dfrac35}=-\dfrac43$

Because the argument's cosine is negative and its sine is positive, you know that $\dfrac\pi2$ . This is important to know because it's only the case that $y=\tan x\implies \arctan y=x$ whenever $-\dfrac\pi2$ . The inverse doesn't exist otherwise.

However, you can restrict the domain of the tangent function so that an inverse can be defined. By shifting the argument of tangent by $\pi$ , we have

$\tan(x-\pi)=y\implies x=\pi+\arctan y$

All this to say

$\tan x=-\dfrac43\implies x=\pi+\arctan\left(-\dfrac43\right)\approx2.2143\text{ rad}\approx126.9^\circ$

So, $-3+4i=5(\cos126.9^\circ+i\sin126.9^\circ)$ .

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