Question

evaluate the expression given sin u = 5/13 and cos v = -3/5 where angle u is in quadrant 2and angle v is in quadrant 2sin ( u - v )

Answer 1

We are given the following information

sin u = 5/13

cos v = -3/5

Where the angle u and v are in the 2nd quadrant.

`\begin{gathered} \cos\theta=(adjacent)/(hypotenuse) \\ \sin\theta=(opposite)/(hypotenuse) \end{gathered}`

Let us find cos u

Apply the Pythagorean theorem to find the 3rd side.

`\begin{gathered} a^2+b^2=c^2 \\ a^2=c^2-b^2 \\ a^2=13^2-5^2 \\ a^2=169-25 \\ a^2=144 \\ a=√(144) \\ a=12 \end{gathered}`

Cos u = 12/13

Now, let us find sin v

Apply the Pythagorean theorem to find the 3rd side.

`\begin{gathered} a^2+b^2=c^2 \\ b^2=c^2-a^2 \\ b^2=5^2-(-3)^2 \\ b^2=25-9 \\ b^2=16 \\ b=√(16) \\ b=4 \end{gathered}`

Sin v = 4/5

Recall the formula for sin (A - B)

`\sin(A-B)=\sin A\cos B-\cos A\sin B`

Let us apply the above formula to the given expression

`\begin{gathered} \sin(u-v)=\sin u\cdot\cos v+\cos u\cdot\sin v \\ \sin(u-v)=(5)/(13)\cdot-(3)/(5)+(12)/(13)\cdot(4)/(5) \\ \sin(u-v)=(33)/(65) \end{gathered}`

Therefore, sin (u - v) = 33/65