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 [tex] \frac{ \sqrt{3} }{2} [/tex], the cos of 45 is [tex] \frac{1}{ \sqrt{2} } [/tex], the cos of 60 is 1/2, and the sin of 45 is [tex] \frac{1}{ \sqrt{2} } [/tex]. When we put all that together we get [tex]( \frac{ \sqrt{3} }{2} * \frac{1}{ \sqrt{2} } )+( \frac{1}{2}* \frac{1}{ \sqrt{2} }) [/tex]. Simplifying all of that we have [tex] \frac{ \sqrt{3} }{2 \sqrt{2} } + \frac{1}{2 \sqrt{2} } [/tex]. We can put that over the common denominator that is already there and get [tex] \frac{1+ \sqrt{3} }{2 \sqrt{2} } [/tex]. 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.
