1) Find the values (if possible) of the six trigonometric functions of o if the terminal side of o lies on the given line in the specified quadrant.

y=1/3x

2)Evaluate (if possible)the sine, cosine, and tangent of the angles without a calculator.

(a) 10pie/3
(b) 17pie/3

1) A y = x/3 line makes an angle of arctan 1/3 with the x axis. That line is in the first quadrant (and also the third quadrant).

You have not specified the quadrant.

tan theta = 1/3, by definition
sin theta = 1/sqrt10 (in the 1st quadrant)
cos theta = 3/sqrt10
etc.

2) (a) Subtract 2 pi from 10 pi/3 you will see that it has the same trig functions as 4 pi/3. The reference angle is pi/3 and it is in the third quadrant. The cosine is -1/2 etc. You should be able to figure out the others.

(b) Subtract 4 pi = 12 pi/3 and you get
5 pi/3 . The reference angle is pi/3 and it is in the fourth quadrant. The cosine is 1/2, etc

To find the values of the trigonometric functions for an angle o that lies on a line in a specific quadrant, we can use the properties of the trigonometric functions and the coordinates of the point on the line where the terminal side of o intersects with the unit circle.

For the first question, we have the equation of the line y = (1/3)x.

1) Determine the quadrant:
To determine the quadrant, we need to look at the signs of x and y.
- In the first quadrant, both x and y are positive.
- In the second quadrant, x is negative, and y is positive.
- In the third quadrant, both x and y are negative.
- In the fourth quadrant, x is positive, and y is negative.

In the case of y = (1/3)x, we can see that both x and y are positive, so the angle o lies in the first quadrant.

2) Find the coordinates:
To find the coordinates of the point where the terminal side of o intersects with the line y = (1/3)x, we can substitute any value of x into the equation and solve for y. Let's take x = 3 (just as an example):

y = (1/3)(3)
y = 1

Thus, the coordinates of the point are (3, 1).

3) Calculate the trigonometric functions:
Now, using the coordinates of the point (3, 1), we can find the values of the six trigonometric functions for angle o.

- Sine (sin o): sin o = y / r, where r is the distance from the origin to the point.
Since the point (3, 1) lies on the unit circle (radius = 1), sin o = y / 1 = 1.

- Cosine (cos o): cos o = x / r.
cos o = 3 / 1 = 3.

- Tangent (tan o): tan o = sin o / cos o.
tan o = 1 / 3.

- Cosecant (csc o): csc o = 1 / sin o.
Since sin o = 1, csc o = 1 / 1 = 1.

- Secant (sec o): sec o = 1 / cos o.
sec o = 1 / 3.

- Cotangent (cot o): cot o = 1 / tan o.
cot o = 3 / 1 = 3.

For the second question, we need to evaluate the sine, cosine, and tangent of the given angles without a calculator.

(a) For the angle 10π/3:
To evaluate the trigonometric functions for this angle, we need to convert it to an angle within one full revolution by subtracting 2π (360 degrees) until we get an angle between 0 and 2π.
10π/3 - 2π = 4π/3

Now, let's calculate the trigonometric functions for the angle 4π/3:

- Sine (sin o): Since sin (4π/3) is negative in the unit circle for 4π/3, the value is -√3/2.
- Cosine (cos o): Since cos (4π/3) is negative in the unit circle for 4π/3, the value is -1/2.
- Tangent (tan o): tan (4π/3) = sin (4π/3) / cos (4π/3) = (-√3/2) / (-1/2) = √3.

(b) For the angle 17π/3:
Again, we convert the angle to an equivalent angle within one full revolution.
17π/3 - 2π = 11π/3

Now, let's calculate the trigonometric functions for the angle 11π/3:

- Sine (sin o): Since sin (11π/3) is positive in the unit circle for 11π/3, the value is √3/2.
- Cosine (cos o): Since cos (11π/3) is negative in the unit circle for 11π/3, the value is -1/2.
- Tangent (tan o): tan (11π/3) = sin (11π/3) / cos (11π/3) = (√3/2) / (-1/2) = -√3.

Thus, for the angles 10π/3 and 17π/3, the values of the sine, cosine, and tangent are as follows:
10π/3: sin (10π/3) = -√3/2, cos (10π/3) = -1/2, tan (10π/3) = √3.
17π/3: sin (17π/3) = √3/2, cos (17π/3) = -1/2, tan (17π/3) = -√3.