The terminal side of π lies on the given line in the specified quadrant. Find the exact values of the six trigonometric functions of π by finding a point on the line.
y=-x quadrant 2
π = 3Ο/4
sinπ = -1/β2
and so on
To find a point on the line and determine the values of the six trigonometric functions for the angle π in quadrant 2, we can follow these steps:
Step 1: Identify the point on the line.
Since the line equation is y = -x, to find a point on the line, we need to substitute a specific value for x or y. In this case, since we're interested in quadrant 2 where x is negative, we'll choose x = -1.
Substituting x = -1 into the equation, we get:
y = -(-1) = 1
So, a point on the line y = -x in quadrant 2 is (-1, 1).
Step 2: Determine the values of the trigonometric functions.
Now that we have a point (-1, 1) on the line, we can use the coordinates of this point to calculate the values of the trigonometric functions.
The angle π is the angle formed between the positive x-axis and the line connecting the origin to the point (-1, 1).
Using the coordinates of the point, we can determine the values as follows:
1. Sine (sin): sin(π) = y / r
In this case, y = 1 and r = β((-1)^2 + 1^2) = β2, which is the distance from the origin to the point (-1, 1).
Thus, sin(π) = 1 / β2 = β2 / 2.
2. Cosine (cos): cos(π) = x / r
In this case, x = -1 and r = β((-1)^2 + 1^2) = β2.
Thus, cos(π) = -1 / β2 = -β2 / 2.
3. Tangent (tan): tan(π) = y / x
In this case, y = 1 and x = -1.
Thus, tan(π) = 1 / -1 = -1.
4. Cosecant (csc): csc(π) = 1 / sin(π)
In this case, since sin(π) = β2 / 2, csc(π) = 1 / (β2 / 2) = 2 / β2 = β2.
5. Secant (sec): sec(π) = 1 / cos(π)
In this case, since cos(π) = -β2 / 2, sec(π) = 1 / (-β2 / 2) = -2 / β2 = -β2.
6. Cotangent (cot): cot(π) = 1 / tan(π)
In this case, since tan(π) = -1, cot(π) = 1 / (-1) = -1.
So, the exact values of the six trigonometric functions of π in quadrant 2 are:
sin(π) = β2 / 2
cos(π) = -β2 / 2
tan(π) = -1
csc(π) = β2
sec(π) = -β2
cot(π) = -1