FInd the exact value pf the 6 trig function of alpha
Given: point (-2,-6) on the terminal side of the angle in standard position
sin alpha
cos alpha
tan alpha
sec alpha
csc alpha
cot alpha
(0,0), (-2,-6).
X = -2-0 = -2.
Y = -6-0 = -6.
r = sqrt((-2)^2+(-6)^2) = sqrt40 =
sqrt(4*10) = 2sqrt10.
sinA = Y/r = -6/2sqrt10 = -3/sqrt10 =
-3sqrt10/10.
cosA = X/r = -2 / 2sqrt10 =
-1 / sqrt10 = -sqrt10 / 10.
tanA = Y/X = -6/-2 = 3.
cscA = 1/sinA = 10 / -3sqrt10 =
10sqrt10 / -3*10 = 10sqrt10 / -30 =
sqrt10 / -3.
secA = 1/cosA = -3 / sqrt10 =
-3sqrt10 / 10.
cotA = 1/tanA = 1/3.
To find the exact values of the 6 trigonometric functions of alpha, we need to determine the values of the opposite, adjacent, and hypotenuse sides of the triangle formed by the point (-2, -6) on the terminal side of the angle in standard position.
First, let's draw a right triangle by connecting the point (-2, -6) with the origin (0, 0).
|
|
(-2, -6) |
| .
| .
|______.
Using the Pythagorean theorem, we can calculate the hypotenuse of the triangle using the coordinates of the point (-2, -6):
hypotenuse = sqrt((-2)^2 + (-6)^2) = sqrt(4 + 36) = sqrt(40) = 2√10
Next, we can determine the values of the opposite and adjacent sides:
- The opposite side is the vertical distance from the point (-2, -6) to the x-axis, which is 6.
- The adjacent side is the horizontal distance from the point (-2, -6) to the y-axis, which is 2.
Now, we can find the values of the trigonometric functions of alpha:
1. sin(alpha) = opposite/hypotenuse = 6/2√10 = (3√10)/√10 = 3
2. cos(alpha) = adjacent/hypotenuse = 2/2√10 = √10
3. tan(alpha) = sin(alpha)/cos(alpha) = 3/√10
4. sec(alpha) = 1/cos(alpha) = 1/√10
5. csc(alpha) = 1/sin(alpha) = 1/3
6. cot(alpha) = 1/tan(alpha) = √10/3
Therefore, the exact values of the 6 trigonometric functions of alpha are:
- sin(alpha) = 3
- cos(alpha) = √10
- tan(alpha) = 3/√10
- sec(alpha) = 1/√10
- csc(alpha) = 1/3
- cot(alpha) = √10/3
To find the exact value of the six trigonometric functions of angle alpha, we can use the given point (-2, -6) on the terminal side of the angle in standard position.
Step 1: Determine the radius.
We can use the distance formula to find the radius (r) of the point (-2,-6) to the origin (0,0).
r = sqrt((-2)^2 + (-6)^2)
r = sqrt(4 + 36)
r = sqrt(40)
r = 2 * sqrt(10)
Step 2: Identify the values of x and y.
The x-coordinate of the given point (-2, -6) represents the value of cosine, and the y-coordinate represents the value of sine.
cos alpha = x / r
cos alpha = -2 / (2 * sqrt(10))
cos alpha = -1 / sqrt(10) [Rationalize the denominator by multiplying both numerator and denominator by sqrt(10)]
sin alpha = y / r
sin alpha = -6 / (2 * sqrt(10))
sin alpha = -3 / sqrt(10) [Rationalize the denominator by multiplying both numerator and denominator by sqrt(10)]
Step 3: Use the values obtained to find the other trigonometric functions.
tan alpha = sin alpha / cos alpha
tan alpha = (-3 / sqrt(10)) / (-1 / sqrt(10))
tan alpha = 3
sec alpha = 1 / cos alpha
sec alpha = 1 / (-1 / sqrt(10))
sec alpha = -sqrt(10)
csc alpha = 1 / sin alpha
csc alpha = 1 / (-3 / sqrt(10))
csc alpha = -sqrt(10) / 3
cot alpha = 1 / tan alpha
cot alpha = 1 / 3
So, the exact values of the six trigonometric functions of angle alpha are:
sin alpha = -3 / sqrt(10)
cos alpha = -1 / sqrt(10)
tan alpha = 3
sec alpha = -sqrt(10)
csc alpha = -sqrt(10) / 3
cot alpha = 1 / 3