log2(-3sinx)=2log2(cosx)+1 solve for x
log 2 (-3sin x) -log2 (cos^2) x =1
log 2 [ -3 sin x/cos^2 x = 1
-3 sin x / cos^2 x = 2^1 = 2
sin x/cos^2 x = -2/3
well the sin must be negative so x is between pi and 2 pi
well try
sin x/(1 - sin^2 x ) = -2/3
let z = sin x
z/(1-z^2) = -2/3
3 z = -2 + 2 z^2
2 z^2 - 3 z - 2 = 0
(2z+1)(z-2) = 0
z = 2, (not possible, sin >1?)
z = -1/2
ah ha
-30 degrees or - pi/6
or pi + pi/6 = 7 pi/6
Cute problem :)
To solve the equation log2(-3sinx) = 2log2(cosx) + 1 for x, we need to simplify the equation and use logarithmic properties.
Step 1: Simplify the equation
Using the property loga(b) = c is equivalent to a^c = b, we can rewrite the equation as follows:
2log2(cosx) = log2(-3sinx) - 1
Step 2: Apply logarithmic properties
By applying the logarithmic property loga(b) - loga(c) = loga(b/c), we can rewrite the equation as:
2log2(cosx) = log2((-3sinx)/2)
Step 3: Convert to exponential form
Rewriting the equation in exponential form, we have:
2^2log2(cosx) = 2^log2((-3sinx)/2)
Simplifying further, we get:
4(cosx) = (-3sinx) / 2
Step 4: Solve for x
To solve for x, we multiply both sides of the equation by 2:
8(cosx) = -3sinx
Expanding further:
8cosx = -3sinx
Now, we divide both sides by cosx:
8 = -3tanx
Dividing both sides by -3 gives:
-8/3 = tanx
Step 5: Find the inverse tangent
To find the value of x, we take the inverse tangent (arctan) of both sides:
x = arctan(-8/3)
The solution for x is x = arctan(-8/3).
To solve the equation log2(-3sinx) = 2log2(cosx) + 1 for x, we need to use some properties of logarithms and algebraic manipulation. Here's how:
Step 1: Using the properties of logarithms, we can rewrite the equation as:
log2(-3sinx) = log2(cosx)^2 + log2(2)
Step 2: Apply the power rule of logarithms to the right side of the equation:
log2(-3sinx) = log2((cosx)^2 * 2)
Step 3: Combine the logarithms on the right side using the sum rule of logarithms:
log2(-3sinx) = log2(2(cosx)^2)
Step 4: Apply the exponential function to both sides of the equation, with base 2, to eliminate the logarithm:
-3sinx = 2(cosx)^2
Step 5: Simplify the equation by multiplying out the right side:
-3sinx = 2cos^2x
Step 6: Rewrite cos^2x using the identity cos^2x = 1 - sin^2x:
-3sinx = 2(1 - sin^2x)
Step 7: Distribute the 2 on the right side of the equation:
-3sinx = 2 - 2sin^2x
Step 8: Rearrange the equation by bringing all terms to one side:
2sin^2x - 3sinx + 2 = 0
Step 9: This is now a quadratic equation in sinx. Solve the quadratic using factoring, completing the square, or the quadratic formula.
After solving the quadratic equation, you will obtain the values of sinx. To find the values of x, you can use the inverse sine function or the unit circle to determine the corresponding angles.