Question

If sin alpha=1/2 andcos theta=1/3,then the values of Alpha + beta will lie in the interval (if theta,alpha are both acute)
(a) [pi/3,pi/2] (b) [pi/2,2pi/3] (c) [2pi/3,5pi/6] (d) [5pi/6,pi]

Answer 1

a option because sinα=1/2 then α=30deg
costheta=1/3 then theta is 50 Deg approximately. then α+theta = 30+50=80deg
so it belongs to( π/3,π/2)





stheta=1/3 then theta belong to 50deg.

Answer 2

answer : option (a) and (b)

given, sinα = 1/2 = p/h .....(1)

here p = 1 and h = 2

from Pythagoras theorem,

h² = p² + b²

so, b = √(2² - 1)² = √3

then, cosα = √3/2 .....(2)

Similarly, cosβ = 1/3 = b/h .....(3)

here, b = 1 and h = 3

from Pythagoras theorem,

h² = b² + p²

p = √(3² - 1) = 2√2

so, sinβ = 2√2/3 .......(4)

now, using formula, sin(A + B) = sinA.cosB + cosA.sinB

so, sin(α + β) = sinα.cosβ + sinβ.cosα

from equations (1), (2), (3) and (4),

= 1/2 × 1/3 + 2√2/3 × √3/2

= (1 + 2√6)/6

= 0.983 [ it is nearest value of 1]

as we know, sine function is increasing function between 0 to pi/2 and decreasing in pi/2 to pi.

at pi/2 value of sine becomes 1

so, getting 0.983 value two times between 0 to pi. one 0 to pi/2 and another pi/2 to pi.

in above options , we can get value of sin(α + β)= 0.983 in [ pi/3, pi/2 ] and [pi/2, 2pi/3 ]

so, options (a) and (b) both are correct.