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Grade 11Trigonometry

my question is-
if in triangle abc, angle c is 120 degrees’
then the value of
cos²A + cos²B – cosA.cosB is {answer is a numerical value}

aryan

8 Years agoGrade 11

1 Answer

Aarushi Ahlawat

8 Years ago

Since A+B+C = 180 degrees in a triangle. Given C=120 degrees. Hence A+B= 60 Degrees. Now solve the given equation to get (A+B) form and put value of the angle.

$cos^2(A)+cos^2(B)-cos(A)cos(B)$

$=cos^2(A)+1-sin^2(B)-cos(A)cos(B)$

$=1+cos^2(A)-sin^2(B)-cos(A)cos(B)$

$=1+cos(A+B)cos(A-B)-cos(A)cos(B)$

= $=1+cos(60^o)\left (cos(A)cos(B)+sin(A)sin(B) \right )-cos(A)cos(B)$

$=1+\frac{1}{2}\left (cos(A)cos(B)+sin(A)sin(B) \right )-cos(A)cos(B)$

$=1-\frac{1}{2}\left (cos(A)cos(B)-sin(A)sin(B) \right )$

$=1-\frac{1}{2}cos(A+B)$

$=1-\frac{1}{2}\frac{1}{2}$

$=\frac{3}{4}$

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