Question

Use the given information to find the exact value of the expression

\(\cos \alpha=-\frac{7}{25}, \alpha\) lies in quadrant \(I I I\), and \(\sin \beta=\frac{\sqrt{21}}{5}, \beta\) lies in quadrant II . Find \(\cos (\alpha+\beta)\).

A. \(\frac{-14-24 \sqrt{21}}{125}\)

B. \(\frac{-48+7 \sqrt{21}}{125}\)

c. \(\frac{48-7 \sqrt{21}}{125}\)

D. \(\frac{14+24 \sqrt{21}}{125}\)

Answer 1

Solution given: Losa=-7 25 a lies in quadrant III and sinß= √21 5 ß lies in quadrant I cos (2+ BJ - los a los B sind sing [:cs (a+b) = cosa cobb- sina sink] -> Sing=f21 A 5 5 31 8 2 use py thagaras the arem BC - / (A)?- (AB)? (15)?- (12172 25-21 cos B 9 s [ B lies in quadrant II] -> A los a =-7 25 25 24 AB= Nincs? - (BL) 2 8 7 C √ (25] 2 (7,2 (625-49 √576 A B = 24

Sina= -24 25 [ a lies in quadrant I ] -> los (&+B) = cosa cosß- sina sinß tase) - ( ) 14 125 + 24/21 125 los (&+B) - 14+24/21 125 D option is correct

Answer 2

ANSWER :

Using a for alpha and b for beta.

cos a = - 7/25 and sin b = sqrt(21) / 5 . (a in quadrant III and b in quadrant II)

=> cos^2 a = (-7/25)^2 = 49/625 

=> sin a = - sqrt( 1 - 49/625)  = - 24/25 (negative value being in III quadrant)

=> sin^2 b = (sqrt(21) / 5)^2 = 21/25

=> cos b = sqrt(1 - 21/25) = - 2/5  (negative value beng in quadrant II)

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

= - 7/25 * (- 2/5)  - (- 24/25) * sqrt(21) / 5)

= 14/125 + 24/25 * sqrt (21) / 5

= (14 + 24 * sqrt(21))  / 125  

 So, Option D is the ANSWER.