If sine theta equals 2/5
and theta is in quadrant II,
find the following trigonometric function values.
Remember sine theta is equal to y divided by r.
Since we are given that sine theta
is equal to 2/5,
which again is equal to y divided by r,
because r is always positive,
we know r is equal to five and y is equal to two.
Let's go ahead and record this.
But we're also told that theta is in quadrant II,
and this is quadrant II.
So let's go ahead and sketch the reference triangle
for angle theta in the second quadrant.
Let's go ahead and assume this is the terminal side
of our angle theta,
and therefore the reference angle is this angle here,
formed by the negative x-axis and the terminal side.
So let's call this theta reference.
Let's also sketch the reference triangle.
So let's go ahead and let this be the reference triangle.
And because sine theta is equal to 2/5,
we know the opposite side, or y, is equal to two,
and the hypotenuse, or r, is equal to five.
In order to find all the trig function values,
we will also need to determine x,
which notice how because we are on the second quadrant
will be negative.
To determine the value of x,
we will use the Pythagorean theorem, which is also
the equation x squared plus y squared equals r squared.
So using x squared plus y squared equals r squared,
we substitute two for y and five for r,
which gives us x squared
plus the square of two
equals the square of five.
Simplifying, we have x squared
plus four equals 25.
Isolating x squared, we subtract four on both sides,
which gives us x squared equals 21.
And now we take the square root
of both sides of the equation.
And when 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 we know x is negative
because we are in the second quadrant,
and therefore we now know x is equal
to negative square root of 21.
Let's also label this on the reference triangle.
And now we have all the information we need
in order to find these five trig function values.
We can either use the definitions of the trig functions
in terms of x, y, and r,
or we can use the reference triangle.
We will discuss both.
To begin, because we know sine theta is equal to 2/5
and sine and cosecant are reciprocals of one another,
and therefore cosecant theta
is equal to the reciprocal of 2/5, which is five halves.
Next, cosine theta is equal to x divided by r,
which is equal to negative square root 21 divided by five.
Or using the reference triangle, we would have the ratio
of the adjacent side to the hypotenuse,
which is also negative square root 21 divided by five.
Cosine theta and secant theta
are reciprocals of one another, and therefore secant theta
is equal to the reciprocal of the cosine function value,
which is five divided by negative square root 21,
or just negative five divided by the square root of 21.
Next, we have tangent theta,
which is equal to y divided by x,
and therefore tangent theta is equal to two
divided by negative square root 21,
or just negative two divided by square root 21.
Or using the reference triangle,
we have the ratio of the opposite side to the adjacent side,
which still gives us two divided by square root negative 21,
or negative two divided by square root 21.
And tangent theta and cotangent theta
are reciprocals of one another,
and therefore cotangent theta
is equal to negative square root 21 divided by two.
Just in case you are asked to rationalize the denominators,
let's rationalize the denominator
of the tangent function value,
as well as the secant function value.
To rationalize the denominator of negative two
divided by square root 21,
we multiply the numerator and denominator
by the square root of 21,
which gives us negative two square root 21 divided by 21.
Similarly, if we rationalize negative five
divided by square root 21,
we get negative five square root 21 divided by 21.
I hope you found this helpful.