solve
4 sin x = cos x for 0<X<360
divide both sides by cosx
4 sinx/cosx = 1
4 tanx = 1
tanx = 1/4
x = arctan(1/4) = 14.04°
but the tangent is also positive in quad III
so x = 180+14.04 = 194.04°
x = 14.04° or 194.04°
how do i know if the tangent is positive in the third quadrant?
To solve the equation 4sin(x) = cos(x) for 0 < x < 360, we can use trigonometric identities and algebraic manipulations.
Let's start by rearranging the equation to isolate the cosine term:
cos(x) = 4sin(x)
Next, we can use the Pythagorean identity sin²(x) + cos²(x) = 1 to replace sin²(x) in terms of cos(x):
1 - cos²(x) + cos²(x) = 1
1 = 1
Now, substitute 4sin(x) with 4√(1 - cos²(x)):
cos(x) = 4√(1 - cos²(x))
Squaring both sides of the equation:
cos²(x) = 16(1 - cos²(x))
cos²(x) = 16 - 16cos²(x)
17cos²(x) = 16
cos²(x) = 16/17
cos(x) = ± √(16/17)
Now, we have two possibilities for cos(x):
cos(x) = √(16/17) or cos(x) = -√(16/17)
To find the corresponding values of x, we can take the inverse cosine (arccos) of each possibility:
For cos(x) = √(16/17):
x = arccos(√(16/17))
For cos(x) = -√(16/17):
x = arccos(-√(16/17))
Finally, calculate the value of x in the given range (0 < x < 360) by using a calculator or by working with the unit circle.