SOLUTION: find the exact value given the following conditions a. cos (a + b) b. sin (a + b) c. tan (a + b) sin a = 5/6, pi/2 < a < pi, and tan b = 3/7, pi < b < 3pi/2

Algebra ->  Trigonometry-basics -> SOLUTION: find the exact value given the following conditions a. cos (a + b) b. sin (a + b) c. tan (a + b) sin a = 5/6, pi/2 < a < pi, and tan b = 3/7, pi < b < 3pi/2      Log On


   

Question 633150: find the exact value given the following conditions
a. cos (a + b)
b. sin (a + b)
c. tan (a + b)
sin a = 5/6, pi/2 < a < pi, and tan b = 3/7, pi < b < 3pi/2
Answer by lwsshak3(11628) About Me  (Show Source):
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find the exact value given the following conditions
a. cos (a + b)
b. sin (a + b)
c. tan (a + b)
sin a = 5/6, pi/2 < a < pi, and tan b = 3/7, pi < b < 3pi/2
**
O=opposite side
A=adjacent side
H=hypotenuse
..
a is a reference angle in quadrant II where sin>0 and cos<0
sin a=5/6=O/H (given)
A=√(H^2-O^2)=√(36-25)=√11
cos a=-√11/6
tan a=-5/√11
..
b is a reference angle in quadrant III where both sin and cos<0
tan b=3/7=O/A (given)
H=√(O^2+A^2)=√(9+49)=√58
sin b=–3/√58
cos b=–7/√58
..
a. cos (a + b)
=cos a*cos b-sin a*sin b
=[-√11/6*-7/√58]-[(5/6)*(-3√58)]=(7√11+15)/(6√58)
..
b. sin (a + b)
=sin a*cos b+cos a*sin b
=[(5/6)*(-7/√58)+[(-√11/6)*(-3/√58)]
=(-35+3√11)/(6√58)
..
tan(a + b)
=(tan a+tan b)/(1-tan a tan b)
=[(-5/√11)+(3/7)]/[1-(-5/√11)(3/7)]