Question

Find the exact value of sin(u+v) given that sin u=7/25 and cos v=-12/13

Answer 1

Answer:

Step-by-step explanation:

The sum identity for sin(u + v) is:

sin(u + v) = sin(u)cos(v) + cos(u)sin(v)

We were given the value for sin(u) and for cos(v).  But if you notice in the identity, we need sin(v) and cos(u).  We will find those and then fit them into the identity and solve.  First for the cos(u):

The sin ratio is the side opposite the reference angle over the hypotenuse in a right triangle.  If we set the reference angle in standard position (at the origin), and we are in QI, then the side opposite the reference angle is 7 and the hypotenuse is 25, so we need to find the side adjacent to the reference angle.  We will do that using Pythagorean's Theorem:

$7^2+x^2=25^2$ and

$x^2=25^2-7^2$ and

$x^2=625-49$ and

$x^2=576$ so taking the square root of both sides gives us that

x = 24.

If x = 24 and that is the side adjacent to the reference angle, then

$cos(u)=\frac{24}{25}$

We will do the same to find sin(v).  This time we have to be in a quadrant where x is negative.  I set the reference angle in QII since x is negative there.  The reference angle is sitting at the origin, the side adjacent to the reference angle is -12, the hypotenuse is 13, so to find y (the side opposite the reference angle, we will again use Pythagorean's Theorem:

$(-12)^2+y^2=13^2$ and

$y^2=13^2-(-12)^2$ and

$y^2=169-144$ and

$y^2=25$ so

y = 5 and that is the side across from the reference angle.  That means that

$sin(v)=\frac{5}{13}$

Now we have all the identities we need:

$sin(u)=\frac{7}{25}$ , $cos(u)=\frac{24}{25}$ , $sin(v)=\frac{5}{13}$ , $cos(v)=-\frac{12}{13}$

Filling in the formula:

$(\frac{7}{25}*-\frac{12}{13})+(\frac{24}{25}*\frac{5}{13})$ which simplifies to

$-\frac{84}{325}+\frac{120}{325}=\frac{36}{325}$

So there you go!

Answer 2

Answer

Step-by-step explanation: