solve the equation 2 sec x- tan x = cot x for the interval 0 to 360
To solve the equation 2sec(x) - tan(x) = cot(x) for the interval 0 to 360, we'll need to use trigonometric identities and algebraic manipulation.
Step 1: Simplify the trigonometric expressions
Recall the following identities:
- sec(x) = 1/cos(x)
- tan(x) = sin(x)/cos(x)
- cot(x) = 1/tan(x) = cos(x)/sin(x)
Substituting these identities into the equation, we have:
2(1/cos(x)) - (sin(x)/cos(x)) = cos(x)/sin(x)
Step 2: Clear the fractions
To eliminate the denominators, we multiply both sides of the equation by cos(x)sin(x):
2sin(x) - sin(x)cos(x) = cos^2(x)
Step 3: Simplify the equation
Rearrange the equation:
2sin(x) = sin(x)cos(x) + cos^2(x)
Combine like terms:
0 = sin(x)cos(x) + cos^2(x) - 2sin(x)
Step 4: Use trigonometric identities and algebraic manipulation
Apply the Pythagorean Identity: sin^2(x) + cos^2(x) = 1
cos^2(x) = 1 - sin^2(x)
Substitute this result into the equation:
0 = sin(x)cos(x) + (1 - sin^2(x)) - 2sin(x)
Simplify further:
0 = sin(x)cos(x) + 1 - sin^2(x) - 2sin(x)
Rearrange and combine like terms:
0 = -sin^2(x) - sin(x) + sin(x)cos(x) + 1
Step 5: Factor the equation
Rearrange the equation:
0 = -sin^2(x) + sin(x)(cos(x) - 1) + 1
Factor the equation:
0 = (1 - sin(x))(1 + sin(x)) + sin(x)(cos(x) - 1)
The equation can be rewritten as:
0 = (1 + sin(x))(1 - sin(x) + cos(x) - 1)
Simplify further:
0 = sin(x) + cos(x)
Step 6: Determine the solutions
To find solutions, we need to analyze the equation sin(x) + cos(x) = 0.
There are two possibilities for this equation to be true:
1) When sin(x) = 0 and cos(x) = 0
2) When sin(x) is not equal to 0 and cos(x) = -sin(x)
Case 1: sin(x) = 0 and cos(x) = 0
In the interval 0 to 360, this occurs at x = 90 degrees and x = 270 degrees.
Case 2: sin(x) is not equal to 0 and cos(x) = -sin(x)
By substituting cos(x) = -sin(x) into sin(x) + cos(x) = 0, we get:
sin(x) - sin(x) = 0
0 = 0
Since the equation is true for all angles in the interval 0 to 360, any angle except 90 and 270 degrees satisfies sin(x) + cos(x) = 0.
Therefore, the solutions for the equation 2sec(x) - tan(x) = cot(x) in the interval 0 to 360 are x = 90 degrees, x = 270 degrees, and any angle except 90 and 270 degrees.