If tangent theta equals negative 5/3
and sine theta is less than zero,
find the following trigonometric function values.
Since the tangent function value is negative
and tangent theta equals y divided by x,
we know tangent theta is negative in the second quadrant
where x is negative and y is positive,
as well as in the fourth quadrant
where x is positive and y is negative.
But we're also told that sine theta is less than zero,
and since sine theta equals y divided by r,
since r is always positive,
if the sine function value is negative,
we know y must be negative.
And therefore, the terminal side of angle theta must be
in the fourth quadrant where y is negative,
not in the second quadrant where y is positive.
So let's go ahead and sketch the terminal side
of angle theta in the second quadrant.
We know the initial side would be along the positive x-axis.
And again, we now know the terminal side
is in the fourth quadrant.
Let's just say here.
Notice the reference angle
for angle theta would be this angle here.
So let's call this theta reference.
Let's also sketch the reference triangle.
The reference triangle would be this triangle here.
So because we know that tangent theta
is equal to y divided by x
and we know the terminal side is in quadrant four,
let's write the fraction negative 5/3
as negative five divided by a positive three,
which tells us that x is equal to positive three
and y is equal to negative five.
Let's also label these values on the reference triangle
where the opposite side, or y, is negative five
and the adjacent side, or x, is positive three.
And now let's find the hypotenuse or the radius
using the Pythagorean theorem,
or x squared plus y squared equals r squared.
And remember, r is always positive.
So using x squared plus y squared equals r squared,
we substitute positive three for x and negative five for y,
which gives us the square of three
plus the square of negative five equals r squared.
Simplifying, we have nine plus 25 equals r squared.
And therefore, r squared is equal to 34.
And now we take the square root
of both sides of the equation.
Since we know r is positive,
we only take the principal square root.
And therefore, r is equal to the square root of 34.
Let's record this below
and also on the reference triangle as the hypotenuse.
And now we have all the information we need
in order to find these five trig function values.
We can use the definitions here in terms of x, y, and r,
as well as the reference triangle.
Since sine theta is equal to y divided by r,
sine theta is equal to negative five
divided by the square root of 34.
Just in case you are required
to rationalize the denominators,
we'll come back and rationalize this denominator
toward the end of the video.
Using the reference triangle, we would have the ratio
of the opposite side to the hypotenuse,
which gives us the same value.
Since we just found the sine function value,
let's find the cosecant function value,
which is the reciprocal of the sine function value,
or r divided by y.
R divided by y gives us the square root of 34
divided by negative five,
or negative square root of 34 divided by five.
Notice how this is the reciprocal
of the sine function value.
Or using the reference triangle,
we have the ratio of the hypotenuse to the opposite side,
which gives us the same function value.
Cosine theta is equal to x divided by r,
which gives us three divided by the square root of 34.
Or using the reference triangle,
we have the ratio of the adjacent side to the hypotenuse,
which gives us the same value.
Since we just found the cosine function value,
let's find the secant function value
since cosine theta and secant theta
are reciprocals of one another.
Secant theta is equal to r divided by x,
which gives us the square root of 34 divided by three,
which again notice is the reciprocal
of the cosine function value.
Or using the reference triangle,
we have the ratio of the hypotenuse to the adjacent side.
And finally, cotangent theta is equal to x divided by y
or the reciprocal of the tangent function value.
X divided by y is three divided by negative five,
or negative 3/5.
Or using the reference triangle,
we have the ratio of the adjacent side to the opposite side,
which again gives us negative 3/5.
Before we go, let's go ahead and rationalize the denominator
of the sine and cosine function values
just in case you are required to do this.
Let's rationalize the denominator of negative five
divided by the square root of 34.
To do this, we multiply the numerator and denominator
by the square root of 34,
which gives us negative five square root of 34
divided by 34.
So we can also express the sine function value
as negative five square root of 34
divided by 34.
Similarly, we can express the cosine function value
as three square root 34
over 34.
I hope you found this helpful.