To solve the equation 6sin²x + sinxcosx - cos²x = 0, let's first rewrite it in terms of trigonometric identities.
We know that sin²x + cos²x = 1, so we can rewrite the equation as:6(sin²x + cos²x) + sinxcosx - cos²x = 06(1) + sinxcosx - cos²x = 06 + sinxcosx - cos²x = 0
Now, let's rewrite the equation in terms of sine and cosine using the following identities:sinxcosx = 0.5sin(2x)cos²x = 0.5(1 + cos(2x))
The equation becomes:6 + 0.5sin(2x) - 0.5(1 + cos(2x)) = 06 + 0.5sin(2x) - 0.5 - 0.5cos(2x) = 05.5 + 0.5sin(2x) - 0.5cos(2x) = 0
Now, we can use the double angle identities to simplify the equation further:sin(2x) = 2sinxcosxcos(2x) = 2cos²x - 1
Substitute these identities into the equation:5.5 + 0.5(2sinxcosx) - 0.5(2cos²x - 1) = 05.5 + sinxcosx - (cos²x - 0.5) = 05.5 + sinxcosx - cos²x + 0.5 = 05 + sinxcosx - cos²x = 0
So, the simplified equation is now:5 + sinxcosx - cos²x = 0
This is equivalent to the original equation given.
To solve the equation 6sin²x + sinxcosx - cos²x = 0, let's first rewrite it in terms of trigonometric identities.
We know that sin²x + cos²x = 1, so we can rewrite the equation as:
6(sin²x + cos²x) + sinxcosx - cos²x = 0
6(1) + sinxcosx - cos²x = 0
6 + sinxcosx - cos²x = 0
Now, let's rewrite the equation in terms of sine and cosine using the following identities:
sinxcosx = 0.5sin(2x)
cos²x = 0.5(1 + cos(2x))
The equation becomes:
6 + 0.5sin(2x) - 0.5(1 + cos(2x)) = 0
6 + 0.5sin(2x) - 0.5 - 0.5cos(2x) = 0
5.5 + 0.5sin(2x) - 0.5cos(2x) = 0
Now, we can use the double angle identities to simplify the equation further:
sin(2x) = 2sinxcosx
cos(2x) = 2cos²x - 1
Substitute these identities into the equation:
5.5 + 0.5(2sinxcosx) - 0.5(2cos²x - 1) = 0
5.5 + sinxcosx - (cos²x - 0.5) = 0
5.5 + sinxcosx - cos²x + 0.5 = 0
5 + sinxcosx - cos²x = 0
So, the simplified equation is now:
5 + sinxcosx - cos²x = 0
This is equivalent to the original equation given.