How would you find the solutions to this problem?
(cosx)(cscx)=2cosx
(cosx)(cscx)=2cosx
(cosx)(cscx) - 2cosx = 0
cosx(cscx - 2) = 0
cosx = 0 or cscx = 2
x = 90 or 270 or x = 30 or 150 degrees
To find the solutions to the equation (cosx)(cscx) = 2cosx, we can follow these steps:
Step 1: Simplify the expression on the left side.
Recall that cscx is the reciprocal of sinx. Therefore, (cosx)(cscx) is equal to cosx/sinx.
The equation now becomes cosx/sinx = 2cosx.
Step 2: Multiply both sides of the equation by sinx to eliminate the denominator.
cosx/sinx * sinx = 2cosx * sinx.
This simplifies to cosx = 2cosx * sinx.
Step 3: Divide both sides of the equation by cosx.
cosx/cosx = (2cosx * sinx)/cosx.
This simplifies to 1 = 2sinx.
Step 4: Divide both sides of the equation by 2.
1/2 = (2sinx)/2.
This simplifies to 1/2 = sinx.
Step 5: Find the values of x that satisfy the equation sinx = 1/2.
As sinx = 1/2, we can look at the unit circle or use reference angles to find the values of x where sinx equals 1/2.
The solutions for x in radians are x = π/6 + 2πn and x = 5π/6 + 2πn, where n is an integer.
In degrees, the solutions for x are x = 30° + 360°n and x = 150° + 360°n, where n is an integer.
To find the solutions to the equation (cosx)(cscx) = 2cosx, we can follow these steps:
Step 1: Simplify the equation using trigonometric identities:
From the given equation, we know that cscx = 1/sinx. By substituting this value into the equation, we get:
(cosx)(1/sinx) = 2cosx
Now, simplify both sides of the equation:
cosx/sinx = 2cosx
Step 2: Multiply both sides of the equation by sinx to eliminate the denominator:
cosx/sinx * sinx = 2cosx * sinx
This simplifies to:
cosx = 2cosx * sinx
Step 3: Further simplify the equation:
Using the identity sin(2x) = 2sin(x)cos(x), we can rewrite the equation as:
cosx = sin(2x)
Step 4: Solve the equation:
To solve for x, we can take the inverse sine (arcsin) of both sides:
arcsin(cosx) = arcsin(sin(2x))
Since arcsin(sin(2x)) has multiple solutions, we need to consider the different values of x that satisfy the equation.
Step 5: Find the solutions:
By analyzing the unit circle and the values of cosine and sine in each quadrant, we can determine the intervals for x where the equation holds true.
The solutions are in the form of x = nπ ± arcsin(cosx), where n is an integer.
Keep in mind that this is a general approach to solving a trigonometric equation. However, specific techniques may be required based on the form and complexity of the equation.
If you cancel out the cos x on both sides, you get
cscx = 2 = 1/sin x
sinx = 1/2
x = 30 or 150 degrees