find the exact values of THE SIX TRIGONOMETRIC FUNCTIONS OF THE ANGLE THETA WHICH HAS A POINT ON THE TERMINAL SIDE OF (-1,4)
plot (0,0), (-1,0), (0,4) and connect the dots
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(0 , 0), (-1 , 4).
X = -1 - 0 = -1.
Y = 4 - 0 = 4.
r^2 = X^2 + Y^2,
r^2 = (-1)^2 + 4^2 = 17,
r = sqrt(17).
sin(Theta) = Y/r = 4 / sqrt(17) =
4*sqrt(17) / 17.
cos(Theta) = X/r = -1/sqrt(17) =
-1*sqrt(17) / 17.
tan(Theta) = Y/X = 4 / -1 = -4.
csc(Theta) = 1/sin(Theta) = 17 / 4*sqrt(17) = 17*sqrt(17) / 4*17 =
sqrt(17) / 4.
sec(Theta) = r/X = 1/cos(Theta) =
17 / -1*sqrt(17) = -17*sqrt(17) / 17
= -1*sqrt(17).
cot(Theta) = 1/tan(Theta) = X/Y =
-1/4 = -(1/4).
To find the six trigonometric functions of an angle, we need its coordinates on the unit circle.
Given the point (-1, 4) on the terminal side of theta, we can use the Pythagorean theorem to find the hypotenuse of the right triangle.
The hypotenuse, r, is given by the formula: r = sqrt(x^2 + y^2), where x and y are the coordinates of the point.
In this case, x = -1 and y = 4, so r = sqrt((-1)^2 + 4^2) = sqrt(1 + 16) = sqrt(17).
Next, we can determine the values of the trigonometric functions:
1. Sine (sin): sin(theta) = y/r = 4/sqrt(17)
2. Cosine (cos): cos(theta) = x/r = -1/sqrt(17)
3. Tangent (tan): tan(theta) = y/x = 4/-1 = -4
4. Secant (sec): sec(theta) = 1/cos(theta) = 1/(-1/sqrt(17)) = -sqrt(17)
5. Cosecant (csc): csc(theta) = 1/sin(theta) = 1/(4/sqrt(17)) = sqrt(17)/4
6. Cotangent (cot): cot(theta) = 1/tan(theta) = 1/-4 = -1/4
Therefore, the exact values of the six trigonometric functions of the angle theta are:
sin(theta) = 4/sqrt(17),
cos(theta) = -1/sqrt(17),
tan(theta) = -4,
sec(theta) = -sqrt(17),
csc(theta) = sqrt(17)/4, and
cot(theta) = -1/4.
To find the exact values of the six trigonometric functions of an angle, we first need to determine which quadrant the angle lies in by looking at the given point.
The point (-1,4) lies in the second quadrant because the x-coordinate is negative (-1) and the y-coordinate is positive (4).
Next, we can find the values of the trigonometric functions.
1. Sine (sin): The sine function is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. In the second quadrant, the sine function is positive. We can find the length of the opposite side by using the Pythagorean theorem.
In this case, the length of the opposite side is 4, and the hypotenuse can be found using the distance formula:
hypotenuse = √((-1)^2 + 4^2) = √(1 + 16) = √17
Therefore, sin(theta) = 4/√17.
2. Cosine (cos): The cosine function is defined as the ratio of the length of the adjacent side to the hypotenuse in a right triangle. In the second quadrant, the cosine function is negative. We can find the length of the adjacent side using the distance formula.
In this case, the length of the adjacent side is -1.
Therefore, cos(theta) = -1/√17.
3. Tangent (tan): The tangent function is defined as the ratio of the sine to the cosine of an angle. We can use the values we found for sine and cosine to find the tangent.
Therefore, tan(theta) = (4/√17) / (-1/√17) = -4.
4. Cosecant (csc): The cosecant function is defined as the reciprocal of the sine function.
Therefore, csc(theta) = 1/(4/√17) = √17/4.
5. Secant (sec): The secant function is defined as the reciprocal of the cosine function.
Therefore, sec(theta) = 1/(-1/√17) = -√17/1 = -√17.
6. Cotangent (cot): The cotangent function is defined as the reciprocal of the tangent function.
Therefore, cot(theta) = 1/(-4) = -1/4.
So, the exact values of the six trigonometric functions of the angle theta are:
sin(theta) = 4/√17
cos(theta) = -1/√17
tan(theta) = -4
csc(theta) = √17/4
sec(theta) = -√17
cot(theta) = -1/4.