Question

Evaluate csc(90degrees - theta) given costheta equals 4/5

Answer 1

To evaluate csc(90 degrees - theta), we first find sin(theta) using the Pythagorean Identity. Then, we find csc(90 degrees - theta) by taking the reciprocal of sin(theta).

Step-by-step explanation:

We are given that cos(theta) = 4/5. To evaluate csc(90 degrees - theta), we need to find the sine of the angle (90 degrees - theta). Since sine and cosine are related by the Pythagorean Identity, sin^2(theta) + cos^2(theta) = 1, we can find sin(theta) by substituting the given value of cos(theta) into the equation and solving for sin(theta). Once we have sin(theta), we can find csc(90 degrees - theta) using the reciprocal relationship between sine and cosecant.

1. Let's find sin(theta) using the Pythagorean Identity.
   
   Given that cos(theta) = 4/5, we can write:
   
   sin(theta)^2 + (4/5)^2 = 1
   
   Simplifying, we get:
   
   sin(theta)^2 + 16/25 = 1
   
   Subtracting 16/25 from both sides, we have:
   
   sin(theta)^2 = 1 - 16/25 = 9/25
   
   Taking the square root of both sides, we find:
   
   sin(theta) = sqrt(9/25) = 3/5
   
   So sin(theta) = 3/5.
   
   
2. Now, let's find csc(90 degrees - theta). Since csc(theta) is the reciprocal of sin(theta), we can write:
   
   csc(90 degrees - theta) = 1/sin(90 degrees - theta)
   
   We know that sin(90 degrees - theta) is equal to cos(theta), so:
   
   csc(90 degrees - theta) = 1/cos(theta)
   
   Substituting the given value of cos(theta) = 4/5, we find:
   
   csc(90 degrees - theta) = 1/(4/5)
   
   To divide by a fraction, we can invert and multiply, so:
   
   csc(90 degrees - theta) = 1 * (5/4)
   
   Finally, simplifying, we get:
   
   csc(90 degrees - theta) = 5/4.