Given theta equals 315 degrees,
determine the following,
we first want to find the reference angle
and then determine the quadrant
of the terminal side of the angle.
Let's first sketch the angle in standard position
where the initial side is along the positive x-axis
and now because the angle is positive,
we rotate counterclockwise 315 degrees.
So we have 90 degrees, 180 degrees, 270 degrees,
we need to rotate another 45 degrees to reach 315 degrees
and therefore the terminal side of 315 degrees
is here in the fourth quadrant.
The reference angle is the acute angle
formed by the terminal side and the x-axis,
or in this case the positive x-axis, this angle here,
so because one complete rotation
counterclockwise is 360 degrees,
because 360 degrees minus 315 degrees is 45 degrees,
the reference angle is 45 degrees, this angle here.
So now we know the reference angle is 45 degrees
and the terminal side is in this quadrant here,
which is quadrant four,
remember the quadrants are quadrant one, quadrant two,
quadrant three and quadrant four.
And then to find the six trig function values,
let's first find them using a reference triangle
and then we'll also find them using the unit circle.
So let's go ahead and sketch a reference triangle,
which will be this triangle here,
notice how we have a 45, 45 90 right triangle,
looking at our notes below,
that means the legs have a length of one
and the hypotenuse has a length of square root two,
but because we're in the fourth quadrant,
where x is positive and y is negative,
we will label this leg positive one, because x is positive,
this leg negative one, because y is negative
and the hypotenuse is always positive,
which in this case is square root two.
Let's use this reference triangle
to find the six trig function values.
Sine 315 degrees is equal to the ratio
of the opposite side of the hypotenuse,
which is negative one divided by square root two,
we will come back later and rationalize the denominator.
From here, remember sine and cosecant
are reciprocals of one another
and therefore cosecant 315 degrees is equal
to the reciprocal of negative one over square root two,
which is square root two over negative one,
which is just negative square root two.
Cosine 315 degrees is equal to the ratio
of the adjacent side to the hypotenuse,
which is positive one divided by square root two
and because secant and cosine
are reciprocals of one another,
secant 315 degrees is equal
to the reciprocal of the cosine function value,
which is square root two over one or just square root two
and tangent 315 degrees is equal to the ratio
of the opposite side to the adjacent side,
which is negative one divided by positive one,
which is equal to negative one
and cotangent and tangent are reciprocals of one another
and therefore cotangent 315 degrees
is equal to the reciprocal of the tangent function value,
which will be one divided by negative one,
which is still negative one.
Before we look at the unit circle,
let's rationalize the denominators
of the sine and cosine function values,
just in case you are required to do this.
If we have negative one divided by square root two,
to rationalize the denominator,
we multiply the numerator and denominator
by the square root of two,
which gives us negative square root two divided by two.
We can also express the sine function value
as negative square root two divided by two,
similarly we can express the cosine function value
as positive square root two divided by two
and now let's also find these trig function values
using the unit circle.
So we already know in standard position
315 degrees has the initial side here
and the terminal side here rotating counterclockwise,
notice how the terminal side
intersects the unit circle at this point here,
where on the unit circle,
x equals cosine data and y equals sine data
and therefore sine 315 degrees is equal to the y-coordinate
of negative square root two divided by two
and because sine and cosecant
are reciprocals of one another,
cosecant 315 degrees is equal
to negative two divided by square root two
and we will come back and rationalize the denominator.
Cosine 315 degrees is equal to the x-coordinate
of square root two divided by two
and cosine and secant are reciprocals of one another
and therefore secant 315 degrees is equal
to positive two divided by square root two
and then finally tangent 315 degrees
is equal to y divided by x,
which gives us negative square root two over two
divided by square root two over two,
which is equal to negative square root two over two
times the reciprocal of square root two over two,
which is times two over square root two.
Simplifying before multiplying,
there's one, two and two here and here
and there's one square root of two
and square root of two here and here,
all this simplifies to just negative one
and cotangent and tangent are reciprocals of one another
and therefore cotangent 315 degrees
is equal to the reciprocal of negative one,
which is still negative one.
So before we go, let's go ahead
and rationalize these two denominators.
For the cosecant function value,
we have negative two divided by square root two,
to rationalize the denominator,
we multiply the numerator and denominator
by the square root of two,
which gives us negative two square root two all over two
and notice how two divided by two simplifies to one
giving us just negative square root two.
Similarly, positive two divided by square root two
simplifies to positive square root two,
notice whether we use a unit circle
or the reference triangle,
we do get the same six trig function values.
I hope you found this helpful.