Question

1)

Find all exact solutions that exist on the interval [0, 21). (Enter your answers as a comma-separated list.) 2 sin(0) --1

2)

Answer 1

1.

2 sin(theta) = -1, in range [0, 2pi)

So,
sin (theta) = -1 / 2,

In range give sine is negative in between 3pi/2 to 2pi
therefore:

sin (theta) = -1 / 2,
sin(theta) = sin(pi + pi / 6) = sin(7pi / 6)
theta = 7pi / 6

sin (theta) = -1 / 2
sin(theta) = sin(2pi - pi / 6) = sin(11pi / 6)
theta = 11pi / 6

Hence theta = 7pi / 6 and 11pi / 6

Q2/

csc(x) sin(x) + sin(x) - 1 / csc(x)

we know that:
csc(x) = 1 / sin(x),
sin(x) = 1 / csc(x)

So the given expression can be written as:
sin(x) * (1 / sin(x)) + sin(x) - sin(x),
(sin(x) / sin(x)) + sin(x) - sin(x),
1 + 0,
1

Hence the solution is 1

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