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2 years ago

[Sin A + Sin 3A + Sin 5A + Sin 7A]÷[Cos A + Cos 3A + Cos 5A + Cos 7A]=Tan 4A

Mathematics

1 answer:

Ainat [17]2 years ago

6 0

Do the following rewrites:

sin(x) = sin(4x - 3x) = sin(4x) cos(3x) - cos(4x) sin(3x)

sin(7x) = sin(4x + 3x) = sin(4x) cos(3x) + cos(4x) sin(3x)

sin(3x) = sin(4x - x) = sin(4x) cos(x) - cos(4x) sin(x)

sin(5x) = sin(4x + x) = sin(4x) cos(x) + cos(4x) sin(x)

cos(x) = cos(4x - 3x) = cos(4x) cos(3x) + sin(4x) sin(3x)

cos(7x) = cos(4x + 3x) = cos(4x) cos(3x) - sin(4x) sin(3x)

cos(3x) = cos(4x - x) = cos(4x) cos(x) + sin(4x) sin(x)

cos(5x) = cos(4x + x) = cos(4x) cos(x) - sin(4x) sin(x)

So in the numerator, we have

sin(x) + sin(3x) + sin(5x) + sin(7x)

= 2 sin(4x) cos(3x) + 2 sin(4x) cos(x)

= 2 sin(4x) (cos(3x) + cos(x))

In the denominator,

cos(x) + cos(3x) + cos(5x) + cos(7x)

= 2 cos(4x) cos(3x) + 2 cos(4x) cos(x)

= 2 cos(4x) (cos(3x) + cos(x))

So we have

(sin(x) + sin(3x) + sin(5x) + sin(7x)) / (cos(x) + cos(3x) + cos(5x) + cos(7x))

= (2 sin(4x) (cos(3x) + cos(x))) / (2 cos(4x) (cos(3x) + cos(x)))

= sin(4x) / cos(4x)

= tan(4x)

QED

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What’s the linear equation for (1,21)(2,33)(3,45)(4,57)

nalin [4]

The linear equation representing the above said pair of points is "y=12x+9"

Step-by-step explanation:

The given set of points are

x Y

1 21

2 33

3 45

4 57

For finding the linear equation for the given sets of value

We must know the generic form of a linear equation is y=m*x + c

m= slope of the line where y= Δy/Δx

Δy= change in y value

Δx= change in x value

Thus slope ”m” = 33-21/2-1 = 12

we put slope “m” in the equation which becomes y=12x+c

Now we put any of the set value in the equation

33= 12*2+c ∴ c=9

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3 years ago

In circle o,find the value of x, rounded to the nearest tenth.

Gemiola [76]

Answer:

$\huge\boxed{x \approx 5.4}$

Step-by-step explanation:

We can break down this problem by first realizing different parts of the circle.

The line which is 8 units long is a chord of the circle.
The line that is 3.6 is almost the radius of the circle
The line that x sits on is the radius.

With this, we can find out if we find the radius of the circle, we have our answer.

We should also note that the angle formed by the 3.6 units long line and the chord is a right angle.

What we need is a way to find the radius of the circle. This will get us x. The radius of a circle will be the length of any line that starts from point O and ends at the circle edge.

If we draw a line connecting the end of the 3.6 line at point O to the end of the 8 unit long chord, we get a triangle! (Image attached for reference).

We can solve for the hypotenuse using the Pythagorean Theorem. This theorem states that:

$\displaystyle a^2 + b^2 = c^2$

Since we know one side is 3.6, we can use that as A. The second side will be 4 since the 3.6 line lies directly in the center of the chord = 8/2 = 4!

$3.6^2 + 4^2 = c^2$
$12.96+16=c^2$
$28.96=c^2$
$\sqrt{28.96} = c$
$5.4 \approx c$

Therefore, since this is the radius of the circle (also the hypotenuse), this can be said for any line that comes from point O onto the edge of the circle.

The line X does just that. Therefore, the value of x is also 5.4.

Hope this helped!