#1**+3 **

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.

hectictar
Apr 15, 2017

#2**+2 **

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} + sin^{2} θ = 1^{2}

sin^{2} 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.

\(-\frac45\cdot\frac43=-\frac{16}{15}\)

hectictar
Apr 15, 2017

#3**+2 **

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..

\(\csc(\arcsin(\frac34))=\frac{1}{\sin(\arcsin(\frac34))}=\frac{1}{\frac34}=1\cdot\frac43=\frac43\)

hectictar
Apr 15, 2017