Graphing Trig. Functions & Inverses

Question

Evaluate In degrees:

 

1. Sin^-1 (-2pi) 

 

2. arctan (- square root 3/3)

 

3. sin (cos^-1(- square root 3/2) 

 

 

Evaluate in Radians

 

1. Arctan (-1)

 

2. cos^-1 (- 1/2) 

Answer 1

Hi Will.

 

Degrees

 

1. sin^-1(-2pi) is difficult to evaluate, because -2pi is generally seen as an angle. Inverse trig functions take ratios as their input and have angles as their outputs. When you input an angle to an inverse trig function you'll need a calculator to evaluate. If, however, you meant to say sin(-2pi), this is the same as -sin(2pi) = 0, 360.

 

2. arctan(-√3/3); in this scenario you want to find the angle which makes tan(x) = -√3/3. Since tangent is y/x, it is negative in both the 2nd and 4th quadrant, so the angles we're looking for will lie somewhere in them. If you look at the ratios of a 30-60-90 triangle, you'll see that tan(30) = 1/√3, or √3/3. With 30 as your reference angle, you can find the 2nd and 4th quadrant equivalents:

 

II: theta = 180 - reference = 180 - 30  = 150 deg

IV: theta = 360 - reference = 360 - 30 = 330 deg

 

arctan(-√3/3) = 150 deg, 330 deg

 

3. sin(cos^-1(-√3/2)); in this scenario, you want to find the angle that makes cos(x) = -√3/2, and then evaluate the sin of that angle. Cosine is negative in the 2nd and 3rd quadrants. By looking at the 30-60-90 triangle, we find that cos(30) = √3/2. With 30 as your reference, you can determine the 2nd and 3rd quadrant equivalents:

 

II: theta = 180 - reference = 180 - 30 = 150 deg

III: theta = 180 + reference = 180 + 30 = 210 deg

 

Now we want to evaluate sin(150) and sin(210). Again, our reference angle is 30 deg, except now the signs will be a little different. Since sine is the y-value of an angle, it is positive in quadrant II and negative in quadrant III.

 

sin(30) = 1/2, so:

 

sin(150) = 1/2,

sin(210) = -1/2

 

sin(cos^-1(-√3/2)) = 1/2, -1/2

 

**

 

Radians

 

1. arctan(-1); just like above, we want to find the angle that makes tan(x) = -1. Looking at a 45-45-90 triangle, we see that tan(pi/4) = 1. Since tangent is negative in quadrant II and quadrant IV, we need to find their equivalent angles with reference angle pi/4 rads.

 

II: theta = pi - pi/4 = 3pi/4

IV: theta = 2pi - pi/4 = 7pi/4

 

Therefore,

 

arctan(-1) = 3pi/4, 7pi/4

 

2. cos^-1(-1/2); again, we want to find the angle that makes cos(x) = 1/2. From our 30-60-90, we see that cos(pi/3) = 1/2. Since cosine is the x value, it will be negative in quadrant II and quadrant III. With a reference angle of pi/3, we get:

 

II: theta = pi - pi/3 = 2pi/3

III: theta = pi + pi/3 = 4pi/3

 

Therefore,

 

cos^-1(-1/2) = 2pi/3, 4pi/3

 

Hope this helps.