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suter [353]
3 years ago
10

Solve the equation for the interval [0, 2pi). 2sin^2x = sin x

Mathematics
1 answer:
notsponge [240]3 years ago
8 0

Answer:

A

Step-by-step explanation:

We want to solve the equation:

2\sin^2(x)=\sin(x)

For the interval [0, 2π).

First, we can move all the terms to one side. Start off by subtracting sin(x) from both sides:

2\sin^2(x)-\sin(x)=0

We can factor:

\sin(x)\left(2\sin(x)-1\right)=0

By the Zero Product Property:

\sin(x)=0\text{ or } 2\sin(x)-1=0

Solve for each case:

\displaystyle \sin(x)=0\text{ or } \sin(x)=\frac{1}{2}

Use the unit circle to solve:

\displaystyle x=\left\{0, \frac{\pi}{6},\frac{5\pi}{6}, \pi\right\}

Hence, our answer is A.

*Please note that we should not simply divide both sides by sin(x) to acquire 2sin(x) = 1. The problem with the operation is that we are dividing by sin(x), yet we do not know what the value of x is. Thus, one or more values of x may result in sin(x) = 0, and we cannot divide by 0. Hence, we are required to subtract and then factor, unless the question specifically states that sin(x) ≠ 0.

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Step-by-step explanation:

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A line passes through (2,4) and (-2,2). find the value of y if (6,y) lies on the same line
katrin2010 [14]

Answer:

y = 6, rendering the coordinate pair: (6,6)

Step-by-step explanation:

We start by writing the equation of the line that passes through two given points on the plane: (x_1,y_1) and (x_2,y_2) beginning with finding the slope of the segment that joints the points using the slope formula: slope=\frac{y_2-y_1}{x_2-x_1}

Let's call (x_1,y_1) = (2,4), and (x_2,y_2) = (-2,2). Then we have the formula for the slope:

slope=\frac{y_2-y_1}{x_2-x_1}=\frac{2-4}{-2-2} =\frac{-2}{-4} =\frac{1}{2}

Now that we have the slope of the line, we can find the actual equation of the line by using one of the given points, and the "point-slope" form of a line with slope "m" and going through a point (x_0,y_0) - which in our case we defie as one of our given points, let's say (2, 4):

y-y_0=m\,(x-x_0)\\y-4=\frac{1}{2} (x-2)\\y-4=\frac{1}{2} x-1\\y=\frac{1}{2} x-1+4\\y=\frac{1}{2} x+3

now we find what is the "y" value in such line that corresponds to an x-value of "6" to complete the coordinate pair (6, ?). For such we simply evaluate the equation above at x = 6:

y=\frac{1}{2} x+3\\y=\frac{1}{2} (6)+3\\y=3+3\\y=6

Therefore, y must be "6".

7 0
3 years ago
Find sin(u/2) if tan(u)=-7.721 and u is in quadrant 4
morpeh [17]

Answer:

.6601

Step-by-step explanation:

to find u, which we assume is an angle of course, we can take arctan of both sides.  so arctan(-7.721) = -1.4420.  this angle is in quadrant 4, but it is not in the range.  In this instance you just go around the circle again, or add 2pi, which gets us 4.8412.  You can check and you get the same answer if you take the tangent of either.

So now we have a number in the appropriate range  in the appropriate quadrant.  That means we have the correct angle.  The question asks what is sine of that angle cut in half.  so sin(4.8412/2), which gets you your answer.  let em know if something didn't make sense.  

7 0
3 years ago
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