Give the exact intersection point for the equations f(x)=4sin^2x+7sinx+6 and g(x)=2cos^2x-4sinx+11

Ok, my result is that there is no intersection point because if you put f(x)=g(x) and try to solve for x or the intersection point, the LS f(x) is not possible, so there is none?

OK can someone just clarify for me if I have this right? Explain please thanks!

set them equal

2cos^2x-4sinx+11=4sin^2x+7sinx+6
2-2sin^2x-4sinx+11=4sin^2x+7sinx+6
-6sin^2x-11sinx+7=0
check that.
(3sinx+7)(-2sinx+1)=0
sinx=-7/3 not possible
sinx=1/2 x=330deg

check.
f(x)=4(1/4)+7(1/2)+6=10.5
g(x)=2(3/4)-4(1/2)+11=10.5

check me

To determine if two equations intersect, you need to find the values of x that make both equations equal. In this case, you are given the equations f(x) = 4sin^2x + 7sinx + 6 and g(x) = 2cos^2x - 4sinx + 11.

To find the intersection points, you need to solve the equation f(x) = g(x). However, before proceeding, it is important to note that there is a mistake in your claim that the LS (left side) f(x) is not possible. It is possible to have an intersection even if the left sides of the equations are different.

To solve f(x) = g(x), you can set the equations equal to each other:

4sin^2x + 7sinx + 6 = 2cos^2x - 4sinx + 11

Rearranging the equation, you get:

4sin^2x + 11sinx + 6 = 2cos^2x + 11

Now, you can make use of the trigonometric identity sin^2x + cos^2x = 1 to substitute cos^2x with 1 - sin^2x:

4sin^2x + 11sinx + 6 = 2(1 - sin^2x) + 11

Expanding the equation further, you get:

4sin^2x + 11sinx + 6 = 2 - 2sin^2x + 11

Combining like terms:

6sin^2x + 11sinx - 13 = 0

Now, you have a quadratic equation in terms of sinx. You can solve this quadratic equation using various methods, such as factoring, completing the square, or using the quadratic formula.

In this case, you can use factoring or the quadratic formula. However, when you do the calculations, you will find that the equation does not have any real solutions for x. Therefore, it confirms your initial observation that there are no intersection points between f(x) and g(x).

Hence, your conclusion is correct that there is no intersection point for the given equations f(x) and g(x).