The given equations are:
1) sin2x - cosx = 2sinx - 12) log3x + 4log9x = 9
Let's solve each equation separately:
1) sin2x - cosx = 2sinx - 1
Using the trigonometric identity sin(2x) = 2sinx*cosx, we can rewrite the equation as:
2sinx*cosx - cosx = 2sinx - 1
cosx(2sinx - 1) = 2sinx - 1
cosx = 1
However, the range of cosine function is [-1, 1]. Therefore, there are no solutions to this equation.
2) log3x + 4log9x = 9
Using the property of logarithms that states logb(x) + logb(y) = logb(xy), we can combine the logarithms:
log3x + log(9x)^4 = 9
log(3x * (9x)^4) = 9
log(3x 9^4 x^4) = 9
log(3 9^4 x^5) = 9
log(3 6561 x^5) = 9
log(19683x^5) = 9
19683x^5 = 10^9
x^5 = 10^9 / 19683
x^5 = 50625
x = 50625^(1/5)
x ≈ 4.764
Therefore, the solution to the second equation is x ≈ 4.764.
The given equations are:
1) sin2x - cosx = 2sinx - 1
2) log3x + 4log9x = 9
Let's solve each equation separately:
1) sin2x - cosx = 2sinx - 1
Using the trigonometric identity sin(2x) = 2sinx*cosx, we can rewrite the equation as:
2sinx*cosx - cosx = 2sinx - 1
cosx(2sinx - 1) = 2sinx - 1
cosx = 1
However, the range of cosine function is [-1, 1]. Therefore, there are no solutions to this equation.
2) log3x + 4log9x = 9
Using the property of logarithms that states logb(x) + logb(y) = logb(xy), we can combine the logarithms:
log3x + log(9x)^4 = 9
log(3x * (9x)^4) = 9
log(3x 9^4 x^4) = 9
log(3 9^4 x^5) = 9
log(3 6561 x^5) = 9
log(19683x^5) = 9
19683x^5 = 10^9
x^5 = 10^9 / 19683
x^5 = 50625
x = 50625^(1/5)
x ≈ 4.764
Therefore, the solution to the second equation is x ≈ 4.764.