(a) sketch an angle A in standard position whose terminal ray passses through the point (1,-4)

(b) Find the exact value of the six trigonometric functions of A,

(c) Let B be the reference angle of A. Find a point on the terminal rsy of B in standard position and add B to your sketch.

construct your right-angled triangle by drawing a perpendicular to the x-axis from (1,-4)

then r^2 = x^2 + y^2 = 1 + 16
r = √17

b)
sinA = -4/√17
cscA = -√17/4

cosA = 1/√17
secA = √17

tanA = -4/1 = -4
cotA = -1/4

(a) To sketch an angle A in standard position whose terminal ray passes through the point (1,-4), we can follow these steps:

1. Plot the point (1, -4) on a coordinate plane.
2. Start at the origin (0,0) and draw a line segment connecting the origin to the plotted point (1, -4).
3. This line segment represents the terminal ray of the angle A.

(b) To find the exact value of the six trigonometric functions of angle A, we need to know the angle measure. Since the angle A is not specified, we can use the coordinates of the point (1, -4) to determine the trigonometric functions.

Let's denote the coordinates of the point (1, -4) as (x, y). From the coordinate plane, we can see that x = 1 and y = -4.

The six trigonometric functions are defined as follows:

1. Sine (sin): sin(A) = y/r, where r represents the distance from the origin to the point (1, -4). Using the Pythagorean theorem, we can calculate r as follows:
r = √(x^2 + y^2)
= √(1^2 + (-4)^2)
= √(1 + 16)
= √17
Thus, sin(A) = y/r = -4/√17.

2. Cosine (cos): cos(A) = x/r = 1/√17.

3. Tangent (tan): tan(A) = sin(A)/cos(A) = (-4/√17)/(1/√17) = -4/1 = -4.

4. Cosecant (csc): csc(A) = 1/sin(A) = 1/(-4/√17) = -√17/4.

5. Secant (sec): sec(A) = 1/cos(A) = 1/(1/√17) = √17.

6. Cotangent (cot): cot(A) = 1/tan(A) = 1/(-4) = -1/4.

Thus, the exact values of the six trigonometric functions of angle A are:
sin(A) = -4/√17, cos(A) = 1/√17, tan(A) = -4, csc(A) = -√17/4, sec(A) = √17, cot(A) = -1/4.

(c) To find the reference angle B of angle A, we need to determine the acute angle formed between the terminal ray of angle A and the x-axis.

Since angle A is in the fourth quadrant (the point (1, -4) is below the x-axis), the reference angle B is the angle formed between the terminal ray of angle A and the positive x-axis.

To find a point on the terminal ray of angle B in standard position, we need to find a point (x, y) that lies on the line segment connecting the origin to the point (1, -4) and has positive y-coordinate.

Let's denote this point as (x, y). Since the line segment crosses the x-axis when y = 0, we can find the x-coordinate by setting y = 0 in the equation of the line connecting the origin to (1, -4).

Using the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept, we can find the equation of the line with points (0,0) (origin) and (1,-4) as follows:

m = (change in y) / (change in x) = (-4 - 0) / (1 - 0) = -4/1 = -4.

Using the point-slope form, we have:
y - y1 = m(x - x1) (where (x1, y1) is a point on the line, which in this case is (0, 0))

y - 0 = -4(x - 0)
y = -4x

Setting y = 0 in the equation, we get:
0 = -4x
x = 0

So the point on the terminal ray of angle B in standard position is (0,0) (origin).

We can then add the angle B to the sketch by drawing a line segment from the origin (0,0) to the point (0,0), which represents the terminal ray of angle B. This line segment coincides with the positive x-axis.

In summary:
(a) Sketch an angle A in standard position whose terminal ray passes through the point (1, -4).
(b) The exact values of the six trigonometric functions of angle A are:
sin(A) = -4/√17
cos(A) = 1/√17
tan(A) = -4
csc(A) = -√17/4
sec(A) = √17
cot(A) = -1/4.
(c) The reference angle B of angle A is 0 degrees, and the point on the terminal ray of angle B in standard position is (0,0) (origin).