If sine theta equals two fifths
and theta is in quadrant two,
find the following trigonometric function values.
Let's begin by sketching a possible terminal side
of angle theta in quadrant two.
Let's just say this is the terminal side of angle theta.
Then the reference angle would be the angle formed
between this ray in the x axis
or this angle here.
Let's also sketch the reference triangle,
which would be this triangle here.
From here, we also know that sine theta is equal
to two fifths and on the coordinate plane,
sine theta is equal to Y divided by R.
And because two fifths is equal to Y divided by R,
we now know that Y is equal to two
and R is equal to five.
But because we want to find these five trig function values,
we also need to determine X.
And we know X is going to be negative
in the second quadrant.
Referring back to the reference triangle.
Because the sine function values equal to two fifths
and the sine function value is equal to the ratio
of the opposite side of the hypotenuse,
we can label the opposite side of the reference angle,
positive two, and the hypotenuse, positive five.
And now, let's find the value of X,
using the Pythagorean theorem,
where the equation X squared
plus Y squared equals R squared.
Using X squared plus Y squared equals R squared,
we substitute two for Y and five for R
and solve for X.
Remember, X is going to be negative.
From the substitution gives us X squared plus the square
of two, equals the square of five.
Simplifying, we have X squared plus four equals 25,
solving for X squared, we subtract four on both sides,
which gives us X squared equals 21.
And now, to solve for X,
we square root both sides of the equation.
And we do this, we include a plus or minus on the right,
which gives us X equals plus or minus the square root of 21.
But again, because we are in the second quadrant,
we know X is going to be negative
and therefore X equals negative square root 21.
And we can also label the adjacent side
in the reference triangle, negative square root 21.
Now we have all the information we need
to determine these five trig function values.
First remember that sine theta
and cosecant theta are reciprocals of one another.
And since sine theta is equal to two fifths,
cosecant theta is equal to the reciprocal of two fifths,
which is five halves.
And now, let's find the cosine function value
as well as the secant function value.
On the coordinate plane,
cosine theta is equal to X divided by R
and therefore cosine theta is equal
to negative square root 21 divided by five.
We're using the reference triangle,
we would have the ratio of the adjacent side
to the hypotenuse,
which also gives us negative square root 21 divided by five.
And the secant function value is equal
to the reciprocal of the cosine function value
and therefore the secant function value
is negative five divided by square root 21.
And now let's find the tangent function value
and the cotangent function value.
Tangent theta is equal Y divided by X,
and therefore tangent theta is equal
to two divided by negative square root 21,
or just negative two over square root 21.
We're using the reference triangle,
we would have the ratio of the opposite side
to the adjacent side,
which still gives us negative two divided by square root 21.
And then the cotangent function value is equal
to the reciprocal of the tangent function value,
and therefore the cotangent function value is equal
to negative square root 21 divided by two.
Before we go,
notice that the denominators of the tangent function value
and the secant function value are irrational.
And if I were to show you how
to rationalize these two trig function values.
If we have negative two divided by square root 21,
to rationalize the denominator,
we multiply the numerator and denominator
by the square root 21.
Multiplying the numerator,
we have negative two, square root 21,
and the denominator, we have square root 21 times itself,
which simplifies perfectly to just 21
and therefore we can also express the tangent function value
as negative two square root 21 divided by 21.
Similarly, for the secant function value,
negative five divided by square root 21 is equal
to negative five square root 21 divided by 21.
I hope you found this helpful.