P is a point on the terminal side of θ in standard position. Find the exact value of the six trigonometric functions for θ. P(-8, -8)

cotangent: 1

secant: negative square root 2
cosecant: negative square root 2
*sin: square root 2 / 2
*cosine: square root 2 / 2
tan: 1

The terminal side lands in the third quadrant, where only tangent (and cotangent) is positive. So sine (cosec) and cosine (sec) are all negative.

Use this as a guide:

S|A
---
T|C

Starting from Q1, we can use
"All Students Take Calculus" to help memorize the positive functions.
(A=all 6 functions)

So I change sin and and cosine to make them negative?

Yes, that is correct, but try to understand why you do that by studying the unit circle.

Thanks! Yes, I will study it.

To find the exact values of the six trigonometric functions for angle θ, we can use the information given about point P(-8, -8) on the terminal side of θ in standard position.

First, let's find the coordinates of point P:
The x-coordinate of point P is -8, and the y-coordinate is -8.

Next, we can use the coordinates of point P to calculate the values of the trigonometric functions.

1. sine (sin):
sin = y/r
sin = (-8) / sqrt((-8)^2 + (-8)^2)
sin = -8 / sqrt(64 + 64)
sin = -8 / sqrt(128)
sin = -8 / (8 * sqrt(2))
sin = -1 / sqrt(2)
To simplify further, we multiply both the numerator and denominator by sqrt(2):
sin = (-1 / sqrt(2)) * (sqrt(2) / sqrt(2))
sin = -sqrt(2) / 2

2. cosine (cos):
cos = x/r
cos = (-8) / sqrt((-8)^2 + (-8)^2)
cos = -8 / sqrt(64 + 64)
cos = -8 / sqrt(128)
cos = -8 / (8 * sqrt(2))
cos = -1 / sqrt(2)
Similar to sin, we can simplify further:
cos = (-1 / sqrt(2)) * (sqrt(2) / sqrt(2))
cos = -sqrt(2) / 2

3. tangent (tan):
tan = y/x
tan = (-8) / (-8)
tan = 1

4. cotangent (cot):
cot = 1 / tan
cot = 1 / 1
cot = 1

5. secant (sec):
sec = 1 / cos
sec = 1 / (-1 / sqrt(2))
sec = -sqrt(2)

6. cosecant (csc):
csc = 1 / sin
csc = 1 / (-1 / sqrt(2))
csc = -sqrt(2)

Therefore, the exact values of the six trigonometric functions for θ are:
sin = -sqrt(2) / 2
cos = -sqrt(2) / 2
tan = 1
cot = 1
sec = -sqrt(2)
csc = -sqrt(2)