Solve the equation 5 sin2x – 7 sinx cosx + 16cos2 x = 4

To solve the equation 5 sin2x – 7 sinx cosx + 16cos2 x = 4, we can follow these steps:

Step 1: Rewrite the equation using trigonometric identities. Using the identity sin2x = 1 - cos2x, we can rewrite the equation as: 5(1 - cos2x) - 7 sinx cosx + 16cos2 x = 4

Step 2: Simplify the equation. Expanding and rearranging the terms, we get: 5 - 5cos2x - 7 sinx cosx + 16cos2 x = 4

Step 3: Combine like terms. Combining the terms with cos2x, we have: -5cos2x + 16cos2 x - 7 sinx cosx = 4 - 5

Simplifying further, we get: 11cos2x - 7 sinx cosx = -1

Step 4: Use the identity cos2x = 1 - sin2x. Substituting cos2x with 1 - sin2x in the equation, we have: 11(1 - sin2x) - 7 sinx cosx = -1

Step 5: Simplify the equation. Expanding and rearranging the terms, we get: 11 - 11sin2x - 7 sinx cosx = -1

Step 6: Combine like terms. Combining the terms with sin2x, we have: -11sin2x - 7 sinx cosx = -1 - 11

Simplifying further, we get: -11sin2x - 7 sinx cosx = -12

Step 7: Factor out sinx. Factoring out sinx, we have: sinx(-11sinx - 7 cosx) = -12

Step 8: Solve for sinx = 0. Setting sinx = 0, we have: 0(-11(0) - 7 cosx) = -12 0 = -12

Since this is not possible, sinx cannot be equal to 0.

Step 9: Solve for -11sinx - 7 cosx = -12. Rearranging the equation, we have: -11sinx - 7 cosx = -12

At this point, we can use various methods such as substitution, elimination, or graphing to solve for sinx and cosx.