Question

Find the exact value of cos(a+b) if cos a=-1/3 and cos b=-1/4 if the terminal side if a lies in quadrant 3 and the terminal side of b lies in quadrant 2

Answer 1

cos(a + b) =
`(1)/(12)(1-2√(30))`

Explanation:

cos(a + b) = cos(a).cos(b) - sin(a).sin(b) [Identity]

cos(a) =
`-(1)/(3)`

cos(b) =
`-(1)/(4)`

Since, terminal side of angle 'a' lies in quadrant 3, sine of angle 'a' will be negative.

sin(a) =
`-\sqrt{1-(-(1)/(3))^2}` [Since, sin(a) =
`\sqrt{(1-\text{cos}^2a)}`]

=
`-\sqrt{(8)/(9)}`

=
`-(2√(2))/(3)`

Similarly, terminal side of angle 'b' lies in quadrant 2, sine of angle 'b' will be negative.

sin(b) =
`-\sqrt{1-(-(1)/(4))^2}`

=
`-\sqrt{(15)/(16)}`

=
`-(√(15))/(4)`

By substituting these values in the identity,

cos(a + b) =
`(-(1)/(3))(-(1)/(4))-(-(2√(2))/(3))(-(√(15))/(4))`

=
`(1)/(12)-(√(120))/(12)`

=
`(1)/(12)(1-√(120))`

=
`(1)/(12)(1-2√(30))`

Therefore, cos(a + b) =
`(1)/(12)(1-2√(30))`