how can i find a general solution of g(x)when they give me g(X) =cos (x+30)and g(X)= -2sinx
well, when does
cos(x+30) = -2sinx
√3/2 cosx - 1/2 sinx = -2sinx
√3/2 cosx = -3/2 sinx
tanx = -1/√3
since tan30 = 1/√3,
x = (180-30) or (360-30) = 150 or 330
or, more generally,
x = 180n-30 for any integer n
To find the general solution of the function g(x) given the equations g(x) = cos(x+30) and g(x) = -2sin(x), we can use trigonometric identities and solve for x.
First, let's start with the equation g(x) = cos(x+30):
Step 1: Use the angle addition formula for cosine: cos(a + b) = cos(a)cos(b) - sin(a)sin(b).
In this case, a = x and b = 30.
cos(x + 30) = cos(x)cos(30) - sin(x)sin(30)
cos(x + 30) = cos(x) * (√3/2) - sin(x) * (1/2)
cos(x + 30) = (√3/2) * cos(x) - (1/2) * sin(x)
Step 2: We know that g(x) = cos(x+30), so substitute the expression we just found:
(√3/2) * cos(x) - (1/2) * sin(x) = cos(x + 30)
Now, let's solve the equation g(x) = -2sin(x):
Step 3: We know that g(x) = -2sin(x), so substitute the expression we have:
-2sin(x) = cos(x + 30)
To find the general solution, we need to solve both equations simultaneously. Since both equations involve trigonometric functions, we can use trigonometric identities to simplify the expressions.
Step 4: Rearrange the equation -2sin(x) = cos(x + 30) to sin(x) by multiplying both sides by -1/2:
sin(x) = -1/2 * cos(x + 30)
Step 5: Use the identity sin^2(x) + cos^2(x) = 1 to eliminate sin(x). Square both sides of the equation sin(x) = -1/2 * cos(x + 30):
sin^2(x) = (-1/2 * cos(x + 30))^2
1 - cos^2(x) = 1/4 * cos^2(x + 30)
Step 6: Multiply both sides of the equation by 4 to eliminate fractions:
4 - 4cos^2(x) = cos^2(x + 30)
Step 7: Combine like terms:
4cos^2(x) + cos^2(x + 30) = 4
Step 8: Apply the double-angle formula for cosine: cos(2a) = 2cos^2(a) - 1.
In this case, a = x:
4cos^2(x) + [2cos^2(x)cos(30) - cos^2(x)sin(30)] = 4
4cos^2(x) + [2cos^2(x) * (√3/2) - cos^2(x) * (1/2)] = 4
4cos^2(x) + (√3cos^2(x) - (cos^2(x)/2)) = 4
4cos^2(x) + (√3/2 * cos^2(x) - cos^2(x)/2) = 4
(9cos^2(x))/2 = 4
9cos^2(x) = 8
cos^2(x) = 8/9
Step 9: Take the square root of both sides:
cos(x) = ±√(8/9)
cos(x) = ±(2√2)/3
Step 10: Solve for x by taking the inverse cosine of both sides:
x = arccos(±(2√2)/3)
The equations x = arccos((2√2)/3) and x = arccos(-(2√2)/3) represent the general solution of g(x) given the two equations. Remember to consider the domain of each equation and apply the appropriate restrictions if necessary.