We're asked to solve the equation,
tangent X equals one for all solutions.
Give degree answers.
Use k to represent any integer.
We will solve this equation first using reference triangles,
then verify the solution using the unit circle,
as well as graphically.
The first thing to notice here is
a tangent function value is positive,
and since tangent theta is equal to Y divided by X
on the coordinate plane, Y divided by X is positive
when both X and Y a positive
or when both X and Y are negative,
which means these solutions will be in the first quadrant
or the third quadrant.
The next step is to find the reference angle that gives
a tangent function value of positive one.
Having a tangent function value of positive one,
should remind us of a 45, 45, 90 reference triangle,
shown here below, where a tangent data is equal to the ratio
of the opposite side to the adjacent side.
Notice tangent 45 degrees is equal to one divided by one,
which is one.
So now we sketch a 45 degree reference angle
in the first quadrant and the third quadrant.
Let's also sketch the reference triangles.
In the first quadrant, both legs are one,
the hypotenuse is square root two.
In the third quadrant, both legs are negative one,
the hypotenuse is square root two.
Notice for both reference triangles,
the ratio of the opposite side of the adjacent side is equal
to the positive one.
And that's what kind of determining all of the solutions.
Let's first focus on the least positive angle
in the first quadrant, which we can see is 45 degrees.
So, we have X equals 45, but we want all the solutions,
and therefore we need an expression for all the angles
that are co-terminal to 45 degrees.
To find co-terminal angles, we add or subtract multiples
of 360 degrees and therefore we have X equals 45 plus 360k,
where k represents any integer,
and the units in this expression would be degrees.
And now let's focus on the least positive angle
in our third quadrant, which should be this angle here.
Notice this angle would be 45 degrees plus 180 degrees,
which is 225 degrees, which gives us X equals 225.
And then for all the co-terminal angles, again we add 360k.
So, while these two expressions do give all the solutions,
there's a simpler way to express these solutions,
since the period of the tangent function is 180 degrees.
Notice how the terminal sides of these angles point
in the opposite direction.
So if we started with 45 degrees
and then added 180 degrees,
we would get the angle of 225 degrees.
If we added 180 degrees again,
we would get another angle that's co-terminal to 45 degrees.
So, a simpler way to express all solutions
for the equation tangent X equals one is,
X equals 45 plus 180k degrees.
So, let's go ahead and give the final solution in this form.
Now let's go to the unit circle and verify the solution.
Since tangent theta is equal to Y divided by X,
and we want Y divided by X to be equal to one,
we're looking for points on the inner circle
where X and Y are equal,
which we can see occur at 45 degrees
where both X and Y are square with two divided by two,
as well as 225 degrees
where both X and Y are negative square two divided by two.
So, the two angles we found using
the reference triangles are correct, but again,
because the period of the tangent function is 180 degrees,
we can't mind all solutions in the simplified expression,
as 45 plus 180K degrees.
And then finally, let's verify some solutions graphically.
In red, we have the graph of Y equals tangent X.
In blue, we have the graph of Y equals one.
The points of intersection represent the solutions
and notice how there would be an infinite number of points
of intersection.
The first angle we found was 45 degrees represented
by this point here.
And if we add multiples of 180 degrees,
we get these solutions in the positive direction.
If we subtract multiples of 180 degrees,
we get solutions in the negative direction.
So, this graph does verify our solution is correct.
I hope you found this helpful.