We're asked to find the exact values
of sine A and cosine A, if tangent A equals negative 5/7
and sine A is greater than zero.
The first step is to find the quadrant
of the terminal side of angle A.
Because we know that tangent A is negative,
and on the coordinate plane, tangent theta is equal
to y divided by x, 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 also know that sine A is greater than 0, or positive.
On the coordinate plane, sine theta is equal to y divided
by r, and because r is always positive,
if the sine function value is positive,
we know y must be positive,
and therefore sine theta is positive in the first quadrant,
where y is positive, as well as in the second quadrant,
where y is positive.
Notice how only quadrant two satisfies the condition
that tangent A is negative and sine A is positive,
which means we know now the terminal side
of angle A must be in quadrant two.
Let's go ahead and sketch the terminal side of angle A,
which again we now know is in quadrant two.
Let's say this is the terminal side of angle A,
and therefore this would be the reference angle for angle A,
and now for the next step, let's go ahead
and sketch the reference triangle for angle A.
The reference triangle would be this triangle here,
and because we know that tangent A is equal to negative 5/7,
which must be equal to y divided by x,
because we're in the second quadrant, where x is negative
and y is positive, let's write negative 5/7
as positive five divided by negative seven,
so now we know that x is equal to negative seven,
and y is equal to positive five,
so going back to the reference triangle,
let's label the opposite side, which is y, positive five,
and the adjacent side, which is x, negative seven,
and the hypotenuse would be r.
Because our goal is to find the exact values
of sine A and cosine A, we now need to find r
using the Pythagorean theorem or the equation
x squared plus y squared equals r squared.
Using x squared plus y squared equals r squared,
we substitute negative seven for x and positive five for y.
This gives us the square of negative seven
plus the square of five equals r squared.
The square of negative seven is positive 49,
plus the square of five is 25.
This gives us 74 equals r squared,
and now to solve for r, we take the square root
of both sides of the equation.
Remember, r must be positive,
so we take the principal square root,
which gives us r equals the square root of 74,
and the square root of 74 does not simplify,
so let's go ahead and replace r with the square root of 74.
Now, let's find the exact value of sine A and cosine A.
Sine A is equal to y divided by r,
which gives us five divided by the square root of 74,
or using the reference triangle,
we have the ratio of the opposite side to the hypotenuse,
which again gives us five divided by the square root of 74.
This is the exact value of sine A, but let's also show
how to rationalize the denominator.
To rationalize the denominator of five divided
by the square root of 74, we multiply the numerator
and denominator by the square root 74.
Multiplying, we have five square root of 74
over the square root of 74 times itself,
simplifies perfectly to just 74.
We can also express the exact sine function value
as five square root of 74 divided by 74.
Cosine A is equal to x divided by r,
which gives us negative seven divided
by the square root of 74, or using the reference triangle,
we have the ratio of the adjacent side to the hypotenuse,
which also gives us negative seven divided
by the square root of 74.
If we rationalize the denominator, this is equal
to negative seven square root of 74 divided by 74.
I hope you found this helpful.