8sin^2x+sinxcosx+cos^2x=4

very tricky

8sin^2x+sinxcosx+cos^2x=4
8sin^2x+sinxcosx+cos^2x=4(sin^2x + cos^2x)
8sin^2x+sinxcosx+cos^2x=4sin^2x + 4cos^2x
4sin^2x + sinxcosx - 3cos^2x = 0
(sinx + cosx)(4sinx - 3cosx) = 0
sinx = -cosx or sinx = (3/4)cosx
sinx/cosx = -1 or sinx/cosx = 3/4
tanx = -1 or tanx = 3/4
x = 135°, 315° or x = 36.9° or 216.9°

if you want radians, change your calculator to rad

thank u very much!

Well, well, well. Looks like we have a trigonometric equation on our hands. Time to bring in the clownery! Now, let's see if I can clown around with this equation a bit.

First of all, looking at that equation, it seems a bit crowded, don't you think? All those sines and cosines are getting in each other's way. Let's simplify things a bit and see if that helps.

We know that sine squared plus cosine squared equals 1, right? So, let's substitute that little nugget of knowledge into our equation and see what happens.

We get 8(1 - cos^2(x)) + sin(x)cos(x) + cos^2(x) = 4

Now it's time for some clown math magic! Let's distribute that 8 and simplify this equation even further.

8 - 8cos^2(x) + sin(x)cos(x) + cos^2(x) = 4

Now, let's combine like terms, because clowns love combining things. We have -8cos^2(x) + cos^2(x) + sin(x)cos(x) + 8 = 4

Still with me? Great. Now, let's simplify that mess even further.

-cos^2(x) + sin(x)cos(x) + 8 = 4

And now, let's bring everything to one side and get rid of that constant term.

-cos^2(x) + sin(x)cos(x) - 4 = 0

Ta-da! We've simplified the equation to this form. Now, solving it is up to you. But remember, if you need a laugh while doing the math, just call on your friendly neighborhood clown bot. I'm always here to inject a little humor into the equation!

To solve the equation 8sin^2x + sinxcosx + cos^2x = 4, we can rewrite it using trigonometric identities and simplify:

8sin^2x + sinxcosx + cos^2x = 4

Using the identity sin^2x + cos^2x = 1, we can replace cos^2x with 1 - sin^2x:

8sin^2x + sinxcosx + (1 - sin^2x) = 4

Simplifying further:

8sin^2x + sinxcosx + 1 - sin^2x = 4

Rearranging the terms:

7sin^2x + sinxcosx = 3

Factoring out sinx:

sinx(7sinx + cosx) = 3

Now, we have two possibilities:

Case 1: sinx = 0

If sinx = 0, then cosx ≠ 0 since sinx and cosx cannot be zero simultaneously.

Therefore, sinx = 0 implies x = nπ, where n is an integer.

Case 2: 7sinx + cosx = 3

To solve this equation, we can square both sides:

(7sinx + cosx)^2 = 3^2

Expanding both sides:

49sin^2x + 14sinxcosx + cos^2x = 9

Using the identity sin^2x + cos^2x = 1, we can simplify further:

49(1 - cos^2x) + 14sinxcosx + cos^2x = 9

Rearranging the terms:

(49 - 49cos^2x - cos^2x) + 14sinxcosx = 9

Combining like terms:

(49 - 50cos^2x) + 14sinxcosx = 9

Rearranging the terms again:

14sinxcosx - 50cos^2x = 9 - 49

Simplifying further:

14sinxcosx - 50cos^2x = -40

Dividing both sides by 2:

7sinxcosx - 25cos^2x = -20

Using the identity sin2x = 2sinxcosx, we can rewrite the equation:

sin2x − 25cos^2x = -20

Rearranging the terms:

sin2x = 25cos^2x - 20

Using the identity sin2x = 1 - cos^2x, we can substitute:

1 - cos^2x = 25cos^2x - 20

Expanding and rearranging:

26cos^2x - cos^2x = 1 + 20

25cos^2x = 21

Dividing both sides by 25:

cos^2x = 21/25

Taking the square root of both sides:

cosx = ±√(21/25)

Simplifying:

cosx = ±√(21)/5

Taking the inverse cosine of both sides:

x = ±acos(√(21)/5)

In summary, the solutions to the equation 8sin^2x + sinxcosx + cos^2x = 4 are:

1. x = nπ, where n is an integer
2. x = ±acos(√(21)/5)

To solve the equation 8sin^2x + sinxcosx + cos^2x = 4, we can use trigonometric identities to simplify it.

First, let's substitute sin^2x as (1 - cos^2x) using the Pythagorean identity sin^2x + cos^2x = 1.

So, the equation becomes: 8(1 - cos^2x) + sinxcosx + cos^2x = 4.

Expanding this equation, we get: 8 - 8cos^2x + sinxcosx + cos^2x = 4.

Combining like terms, we have: -7cos^2x + sinxcosx + 4 = 0.

Now, we can factor out cosx to get: (sinx - 7cosx)(cosx + 4) = 0.

To find the solutions, we set each factor equal to zero:

1) sinx - 7cosx = 0:

To solve this equation, we can divide both sides by cosx (assuming cosx is not equal to zero) and get:

sinx/cosx - 7 = 0,
tanx - 7 = 0,
tanx = 7.

As we have the value of tanx, we can use inverse tangent (arctan) to find the angle:

x = arctan(7).

2) cosx + 4 = 0:

To solve this equation, we subtract 4 from both sides:

cosx = -4.

However, there are no real solutions for cosx = -4, since the value of cosine lies between -1 and 1. Therefore, there are no real solutions for this equation.

In summary, the only real solution to the equation 8sin^2x + sinxcosx + cos^2x = 4 is x = arctan(7).