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.
13)
This one is slightly harder because it has more steps, but it's not too bad!
If cos θ = -3/5 , and sin θ < 0, this is the triangle ∠θ makes:
Use the Pythagorean theorem to find sin θ.
(-3/5)2 + sin2 θ = 12
sin2 θ = 1 - (-3/5)2
Take the negative square root of both sides, since sin θ < 0.
sin θ = \(-\sqrt{1-(-\frac35)^2}=-\sqrt{1-\frac9{25}}=-\sqrt{\frac{16}{25}}=-\frac45\)
By definition of tangent:
tan θ = sin θ / cos θ
So...
tan θ = \(-\frac45 / -\frac35 = \frac45\cdot\frac53=\frac{20}{15}=\frac{4}{3}\)
And so now we have found both tan θ and sin θ.
Just multiply them together to get the final answer.
\(-\frac45\cdot\frac43=-\frac{16}{15}\)
And finally, for this last one!
14)
csc(arcsin (3/4) )
Which is = \(\frac{1}{\sin(\arcsin(\frac34))}\)
Wait a minute, would ya look at that!
What is the sin of an angle that has a sin of 3/4 ??
The sin is 3/4 !
The arcsin(sin (3/4) ) = 3/4
So..
\(\csc(\arcsin(\frac34))=\frac{1}{\sin(\arcsin(\frac34))}=\frac{1}{\frac34}=1\cdot\frac43=\frac43\)