Solve the multiple angle equation in the interval (0, 2ð) tan2x=-1
Do i have to divide the 2 to the other side?
I read you interval as (0 , 2π)
tan 2x = -1
I know tan 45° = tan π/4 = +1
for a negative tangent, the angle is in either II or Iv
so 2x = (180-45)° or 2x = (360-45)°
2x = 135° or 2x = 315° OR 2x = 3π/4 or 2x = 7π/4
x = 67.5° or x = 157.5° OR x = 3π/8 or x = 7π/8
but the period of tan 2x = 90° or π/2
so two more answers in your domain
x = 67.5 + 90 = 157.5 (already have that) or
x = 157.5 + 90 = 247.5°
in radians: x = 7π/8 + π/2 = 11π/8
so:
x = 67.5° , 157.5° , 247.5° OR x = 3π/8 , 7π/8 , 11π/8
Wow, that actually makes perfect sense, thank you!!!!
To solve the equation tan(2x) = -1 in the interval (0, 2π), you need to find the values of x that satisfy the equation. However, before solving, it is not necessary to divide the 2 to the other side.
Here's the step-by-step process to solve this equation:
Step 1: Identify the possible angles satisfying the equation.
Since we are looking for angles in the interval (0, 2π) or (0, 360 degrees), we know that the tangent function is negative in quadrants II and IV. These are the two quadrants where the tangent is negative.
Step 2: Find the reference angle.
The reference angle is the positive acute angle formed between the x-axis and the terminal side of an angle in standard position. For this equation, the reference angle will help us find the general solution.
Step 3: Solve for the reference angle.
To find the reference angle, we can use an inverse trigonometric function. In this case, the reference angle can be found by evaluating arctan(1) or tan^-1(1) on a calculator. The result is π/4 or 45 degrees.
Step 4: Find the general solution.
Since the tangent function has a period of π or 180 degrees, we can add integer multiples of π to the reference angle to find the general solution. In this case, the general solution is given by x = (π/4) + nπ, where n is an integer.
Step 5: Select solutions within the given interval.
Finally, select only those solutions that lie in the interval (0, 2π) or (0, 360 degrees). For this equation, we need to check if the angle (π/4) + nπ lies in the given interval. If it does, then it is a valid solution.
Therefore, if we substitute different integer values for n, we find that x can be:
- x = π/4 when n = 0 (within the given interval)
- x = 5π/4 when n = 1 (within the given interval)
Thus, the solution to the equation tan(2x) = -1 in the interval (0, 2π) is x = π/4 and x = 5π/4.