How do I find all solutions of tan θ= -1?part1 : find the values of θ that satisfy the quation on the interval [0,2pi] part2:write an expression for all solutions to the equation.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find the general solution to equation `tan theta = -1`  such that:

`tan theta = -1 => theta = arctan(-1) + npi`

You need to remember the following property such that:

`arctan(-alpha) = -arctan alpha`

Reasoning by analogy yields:

`theta = -arctan(1) + npi => theta = -pi/4 + npi`

You need to find all solutions of equation `tan theta = -1` , over the interval `[0,2pi]`  hence, you need to remember that the tangent function has negative values in quadrant 2 and 4, thus the equation has two solutions over `[0,2pi]`  such that:

`tan theta = -1 => theta = pi - pi/4 => theta = (3pi)/4`  (quadrant 2)

`tan theta = -1 => theta = 2pi - pi/4 => theta = (7pi)/4`  (quadrant 4)

Hence, evaluating the solutions to the given equation over `[0,2pi]`  yields `theta = (3pi)/4`  and `theta = (7pi)/4`  and evaluating the general solution to the given equation yields `theta = -pi/4 + npi.`

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