Angle A is in quadrant II
and angle B is in quadrant IV.
If sine A equals 2/3 and sine B equals negative 4/5,
determine sine of the quantity A plus B
and sine of the quantity A minus B.
Looking at the sum and difference identities,
notice how we are going to have to find
the cosine function values for angle A and B,
and therefore let's begin by determining
the reference triangles for angle A and angle B.
We know angle A terminates in the second quadrant.
Let's just say this is the terminal side of angle A.
And therefore the reference angle is going to be
this angle here, we'll call it A sub r.
The reference triangle is this right triangle here.
And because we know the sine function value is 2/3,
we can label the opposite side positive two
and the hypotenuse positive three.
And now we can use the Pythagorean Theorem
to determine the length of this leg,
which because we are in the second quadrant,
is going to be negative.
Let's go and label this leg x
and then find x using the Pythagorean Theorem.
Which gives us x squared plus two squared
equals three squared.
Simplifying we have x squared plus four equals nine.
Subtracting four on both sides
gives us x squared equals five.
And now we take the square root of both sides
of the equation, which gives us x equals
plus or minus the square root of five.
But again because we're in the second quadrant,
we know x is negative,
and therefore we label the leg negative square root of five.
We also know angle B is in quadrant IV.
Let's assume this is the terminal side of angle B
in quadrant IV.
The reference angle is going to be this angle here,
we'll call it B sub r.
And the reference triangle is this right triangle here.
Because the sine function value is negative 4/5
and we know the hypotenuse is always positive,
we label the opposite side negative four
and the hypotenuse five.
We also know this leg would be negative
because we're in quadrant IV where y is negative.
And now we need to find this leg here
which we'll label x.
Using the Pythagorean Theorem we have x squared
plus the square of negative four
equals the square of five,
giving us x squared plus 16 equals 25.
Subtracting 16 on both sides gives us
x squared equals nine.
Now we square root both sides of the equation,
which gives us x equals plus or minus three.
But because we're in quadrant IV where x is positive,
this leg is labeled positive three.
Now we have all the information we need to find
sine of the quantity A plus B
and sine of the quantity A minus B.
Using the identities, we have sine
of the quantity A plus B
equals sine A times cosine B plus cosine A times sine B.
Let's go ahead and list out these four trig function
values here below.
Well we know that sine A is 2/3.
Cosine A is equal to the ratio of the adjacent side
to the hypotenuse which is negative square root of five
divided by three.
And then we need to find sine B.
It's given as negative 4/5.
And cosine B
is equal to the ratio of the adjacent side
to the hypotenuse,
which using this reference triangle is 3/5.
And therefore sine of the quantity A plus B
is equal to sine A which is 2/3
times cosine B which is 3/5
plus cosine A which is negative square root five
divided by three times sine B which is negative 4/5.
Simplifying we have 6/15
and then plus a negative times a negative is positive,
and therefore this is plus four square root five fifteenths.
Giving us the quantity six plus four square root of five
all divided by 15.
And now we need to find sine
of the quantity A minus B,
which is equal to sine A times cosine B
minus cosine A times sine B.
Well notice how this expression is the same
as this expression here except instead of a sum
we have a difference,
and therefore this difference is going to be equal to
6/15 minus four square root of five fifteenths
or the quantity six minus four square root five
all divided by 15.
I hope you found this helpful.