Question

Use the sum and difference identities to determine the exact value of the following expression. If the answer is undefined, write DNE.esin 115 cos35 + cos1 15 sin35

Answer 1

Given the following expression;

$\sin 115\cos 35+\cos 115\sin 35$

We begin by using the following trig identity;

$\cos A\sin B+\cos B\sin A=\sin (A+B)$

Using the values of the angles given, we have;

$\begin{gathered} \sin 115\cos 35+\cos 115\sin 35=\sin (115+35) \\ =\sin 150 \end{gathered}$

We can now rewrite as follows;

$\sin 150=\sin (60+90)$

We can now apply the summation identity as identified earlier and we'll have;

$\sin (60+90)=\sin 60\cos 90+\cos 60\sin 90$

At this point we can apply the values of special angles, as shown;

$\begin{gathered} \sin 60=\frac{\sqrt[]{3}}{2},\cos 60=\frac{1}{2} \\ \sin 90=1,\cos 90=0 \end{gathered}$

Substitute these and we now have;

$\begin{gathered} \sin (60+90)=(\frac{\sqrt[]{3}}{2}\times0)+(\frac{1}{2}\times1) \\ \sin (60+90)=0+\frac{1}{2} \\ \sin (60+90)=\frac{1}{2} \end{gathered}$

ANSWER;

The exact value of the given expression is

$\frac{1}{2}$