1. Find cos(a+b) if sin(a)= 8/17 where a is in the 1st quadrant and tan(b)= -7/24 where b is in the 2nd quadrant.

Question

Please Help!!!!!!!

Answer 1

We know that all trig functions are positive in the 1st quadrant, and tangent is negative in 2nd quadrant.  Since angle a is in the 1st quadrant, all trig functions that are in respect to angle a are positive.  Since angle b is in the 2nd quadrant, only sin(b) is positive.

 

We are also given an addition angle identity.

 

cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

 

 

We need to find these 3 values:

 

cos(a) , cos(b), and sin(b)

 

 

We can use the following trig identities when needed to find these missing values:

 

sin²(a) + cos²(a) = 1                   (1)

 

sin²(b) + cos²(b) = 1                   (2)

 

sin(a) / cos(a) = tan(a)                (3)

 

sin(b) / cos(b) = tan(b)                (4)

 

 

Using identity 1, we can find  cos(a).

 

cos(a) = √(1 - sin²(a))

          = √(1 - (8/17)²)

          = √(1 - (64 / 289))

          = √(225 / 289)

          = 15 / 17

 

To find sin(b) and cos(b), we need to find the missing side of the right triangle associated with the tan(b) function.  This is easily found using the Pythagorean theorem.  As you know, tangent is equal to opposite side of angle b over adjacent side of angle b.  So we need to find the hypotenuse, h, of this triangle.

 

h = √((-7)² + 24²)

   = √(49 + 576)

   = √(625)

   = 25

 

Now that we the hypotenuse, we can easily figure out sin(b) and cos(b) using the sides of the triangle.

 

sin(b) = 7 / 25 

 

cos(b) = -24 / 25  

 

 

Now we plug in all of our trig functions to evaluate the addition angle identity.

 

cos(a)cos(b) - sin(a)sin(b) =

 

(15 / 17)(-24 / 25) - (8 / 17)(7 / 25)

 

 

I will let you complete the calculation.