Question

Use an addition or subtraction formula to find the exact value of the expression, as demonstrated in example 1. sin(105°)

Answer 1

105° can be expressed as 60°+45°.  What we have then is sin(60°+45°).  The sum pattern for sin is sin(a)cos(b)+cos(a)sin(b).  We will fill in as follows: sin(60)c0s(45)+cos(60)sin(45).  Now draw those special right triangles in the first quadrant to get the exact values for each.  The sin of 60 is $\frac{ \sqrt{3} }{2}$, the cos of 45 is $\frac{1}{ \sqrt{2} }$, the cos of 60 is 1/2, and the sin of 45 is $\frac{1}{ \sqrt{2} }$.  When we put all that together we get $( \frac{ \sqrt{3} }{2} * \frac{1}{ \sqrt{2} } )+( \frac{1}{2}* \frac{1}{ \sqrt{2} })$.  Simplifying all of that we have $\frac{ \sqrt{3} }{2 \sqrt{2} } + \frac{1}{2 \sqrt{2} }$.  We can put that over the common denominator that is already there and get $\frac{1+ \sqrt{3} }{2 \sqrt{2} }$.  Not sure if that's simplified enough; you may be at the point in class where you are rationalizing your denominator, but I'm not sure, and if you're not, I don't want to confuse you.