arcsin(4/5) is some alpha between -pi/2 and pi/2 with sin alpha = 4/5
arctan(12/5) is some beta between -pi/2 and pi/2 with tan beta = 12/5
We are asked to find sin(alpha + beta). (Do you have some suspicions about why I rewrote the problem this way?)
We know that
sin(alpha + beta) = sin alpha cos beta+cos alpha sinbeta .
We know sin alpha = 4/5 which is positive and with the restriction we already mentioned, we conclude that 0 <= alpha <= pi/2. We want cos alpha.
At this point in our study of trigonometry we have already developed at least one method to find cos alpha (We may have three or four or more methods, but we only need one right now.)
Use your chosen method to get cos alpha = 3/5
So we have:
sin alpha = 4/5" " and " "cos alpha = 3/5
Similarly, given tan beta = 12/5 and 0 <= beta <= pi/2, find sin beta and cos beta
sin beta = 12/13" " and " "cos beta = 5/13
sin(alpha + beta) = sin alpha cos beta+cos alpha sinbeta
= 4/5 5/13 + 3/5 12/13
= 56/65.
If you want to write this without the names alpha and beta 9Or something like that, you could write:
sin(arcsin(4/5)+arctan(12/5))
= sin(arcsin(4/5))cos(arctan(12/5))+ cos(arcsin(4/5))sin(arctan(12/5))
But I, personally do not think that is more clear. (I do think it is good for students to see it this way, however.)
