10)
sin θ = -0.334
Take the arcsin of both sides.
arcsin(sin θ) = arcsin(-0.334)
θ = arcsin(-0.334)
Use a calculator to get that:
θ ≈ -19.512º
But this is not the answer yet. The calculator gives us an angle that is in the fourth quadrant, but θ is in the third quadrant. So, subtract the angle from 180º to find θ.
θ = 180º - (-19.512º )
θ = 180º + 19.512º
θ = 199.5º
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11)
We want another angle that has the same cosine as π / 4 .
This is a picture. The red line is cos π / 4.
As you can see, angle a will have the exact same cosine as π / 4.
π / 4 is 45º above the red line, and angle a is 45º below the red line.
So.. angle a is -45º , or -π / 4 .
But -π / 4 is not in the range of [0 , 2π)
So, just add 360º , or 2π , to get it the right range.
a = -π / 4 + 2π = 7π / 4
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12)
This one isn't too hard.
The tangent values repeat every 180º , or π.
So just add π to 2π / 3 to find another angle with the same tangent.
