Given <a in quadrant 1 and <b in quadrant 2 with sina= 7/25, cosb=-5/13. A) find exact values for cosa and sinb. B) use the appropriate compound angle formula to find the exact value of sin(a+b)

Answer 1

It is given to us that angle α is in the first quadrant and sin α = 7/25; angle β is in the second quadrant and cos β =  – 5/13.

We will be using the property of the trigonometric unit to do the first part of the problem.  

                                               sin² α + cos² α = 1

From this expression cos α = ± √(1 – sin² α) .  

Putting the known values we get that cos α = ± √( 1 – 49/625) = ± 24/25.  Because angle α is in the first quadrant and sin of the angle is positive in this quadrant, we are getting the answer that cos α = 24/25.

Similarly for angle β we have

                                            sin β = ± √(1 – cos² β)

After number substitution we obtain that sin β = ± √(1  – 25/169) = ±  12/13. In the second quadrant the sin of the angle is positive, hence we get sin β = 12/13.

For the second part of the problem we are going to use the formula for the sin of the sum of the angles

                                             sin ( α + β) = sin α∙cos β + cos α∙sin β

Let’s put in this formula the known values

                         sin ( α + β) = (7/25)∙( –5/13) + (24/25)∙(12/13) = 183/325

                                                  Answer: 24/25, 12/13, 183/325



Hope that now you have better understanding of the trigonometry.