Question

you are given that cos(a)=1213, with a in quadrant iv, and cos(b)=513, with b in quadrant i. find cos(a b). give your answer as a fraction.

Answer 1

To find cos(ab), you can use the trigonometric identity cos(ab) = cos(a)cos(b) - sin(a)sin(b). Given cos(a) = 12/13 and cos(b) = 5/13, the value of cos(ab) is 60/169 + (5√39)/169.

Step-by-step explanation:

To find cos(ab), we can use the trigonometric identity:

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

Given that cos(a) = 12/13 and cos(b) = 5/13, we can substitute these values into the identity:

cos(ab) = (12/13)(5/13) - (-√(1 - (12/13)^2))(-√(1 - (5/13)^2))

Simplifying the expression, we get:

cos(ab) = 60/169 + (5√39)/169

Answer 2

To find cos(a + b), use the cosine addition formula cos(a + b) = cos(a)cos(b) - sin(a)sin(b). Calculating sin(a) as -5/13 (since a is in quadrant IV) and sin(b) as 12/13 (since b is in quadrant I), we substitute and find cos(a + b) = 120/169.

Step-by-step explanation:

We are given that cos(a) = 12/13, with angle a in quadrant IV, and cos(b) = 5/13, with angle b in quadrant I. To find cos(a + b), we can use the cosine addition formula:

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

Since a is in quadrant IV, sin(a) is negative. Additionally, since b is in quadrant I, sin(b) is positive. We can use the Pythagorean identity to find sin(a) and sin(b):

sin(a) = -sqrt(1 - cos²(a)) = -sqrt(1 - (12/13)²) = -sqrt(1 - 144/169) = -sqrt(25/169) = -5/13

sin(b) = sqrt(1 - cos²(b)) = sqrt(1 - (5/13)²) = sqrt(1 - 25/169) = sqrt(144/169) = 12/13

Now we substitute the values into the addition formula:

cos(a + b) = (12/13)(5/13) - (-5/13)(12/13)

cos(a + b) = 60/169 - (-60/169)

cos(a + b) = 120/169

Therefore, cos(a + b) = 120/169.