Answer:
cos(a + b) = [tex]\frac{1}{12}(1-2\sqrt{30})[/tex]
Step-by-step explanation:
cos(a + b) = cos(a).cos(b) - sin(a).sin(b) [Identity]
cos(a) = [tex]-\frac{1}{3}[/tex]
cos(b) = [tex]-\frac{1}{4}[/tex]
Since, terminal side of angle 'a' lies in quadrant 3, sine of angle 'a' will be negative.
sin(a) = [tex]-\sqrt{1-(-\frac{1}{3})^2}[/tex] [Since, sin(a) = [tex]\sqrt{(1-\text{cos}^2a)}[/tex]]
= [tex]-\sqrt{\frac{8}{9}}[/tex]
= [tex]-\frac{2\sqrt{2}}{3}[/tex]
Similarly, terminal side of angle 'b' lies in quadrant 2, sine of angle 'b' will be negative.
sin(b) = [tex]-\sqrt{1-(-\frac{1}{4})^2}[/tex]
= [tex]-\sqrt{\frac{15}{16}}[/tex]
= [tex]-\frac{\sqrt{15}}{4}[/tex]
By substituting these values in the identity,
cos(a + b) = [tex](-\frac{1}{3})(-\frac{1}{4})-(-\frac{2\sqrt{2}}{3})(-\frac{\sqrt{15}}{4})[/tex]
= [tex]\frac{1}{12}-\frac{\sqrt{120}}{12}[/tex]
= [tex]\frac{1}{12}(1-\sqrt{120})[/tex]
= [tex]\frac{1}{12}(1-2\sqrt{30})[/tex]
Therefore, cos(a + b) = [tex]\frac{1}{12}(1-2\sqrt{30})[/tex]
