Find the values of the six trigonometric functions of an angle 0 in standard position whose terminal side passes through point P.
P(4, -3)
To find the values of the six trigonometric functions of an angle in standard position whose terminal side passes through point P(4, -3), we can use the coordinates of point P and the Pythagorean theorem.
First, let's find the hypotenuse (r) using the distance formula:
r = √(x² + y²)
r = √(4² + (-3)²)
r = √(16 + 9)
r = √25
r = 5
Now that we have the hypotenuse, we can find the values of the trigonometric functions:
1. The sine of the angle (sinθ) is given by the ratio of the opposite side (y-coordinate) to the hypotenuse:
sinθ = y / r
sinθ = -3 / 5
2. The cosine of the angle (cosθ) is given by the ratio of the adjacent side (x-coordinate) to the hypotenuse:
cosθ = x / r
cosθ = 4 / 5
3. The tangent of the angle (tanθ) is given by the ratio of the opposite side (y-coordinate) to the adjacent side (x-coordinate):
tanθ = y / x
tanθ = -3 / 4
4. The cosecant of the angle (cscθ) is the reciprocal of the sine:
cscθ = 1 / sinθ
cscθ = 1 / (-3 / 5)
cscθ = -5 / 3
5. The secant of the angle (secθ) is the reciprocal of the cosine:
secθ = 1 / cosθ
secθ = 1 / (4 / 5)
secθ = 5 / 4
6. The cotangent of the angle (cotθ) is the reciprocal of the tangent:
cotθ = 1 / tanθ
cotθ = 1 / (-3 / 4)
cotθ = -4 / 3
Therefore, the values of the six trigonometric functions of the angle 0 are as follows:
sinθ = -3 / 5
cosθ = 4 / 5
tanθ = -3 / 4
cscθ = -5 / 3
secθ = 5 / 4
cotθ = -4 / 3
To find the values of the six trigonometric functions of an angle in standard position whose terminal side passes through point P(4, -3), we first need to determine the values of the three sides of the right triangle formed by the terminal side of angle 0 and the x and y axes.
First, let's find the hypotenuse of the triangle using the Pythagorean theorem:
Hypotenuse^2 = x^2 + y^2
Hypotenuse^2 = 4^2 + (-3)^2
Hypotenuse^2 = 16 + 9
Hypotenuse^2 = 25
Hypotenuse = √25 = 5
Next, we can determine the adjacent side of the triangle, which is the x-coordinate of point P(4, -3). The adjacent side represents the distance between the origin and the nearest point on the terminal side of angle 0 along the x-axis. In this case, it is positive 4.
Finally, we need to find the opposite side of the triangle, which is the y-coordinate of point P(4, -3). The opposite side represents the distance between the origin and the nearest point on the terminal side of angle 0 along the y-axis. In this case, it is negative 3.
Now that we know the lengths of all three sides of the right triangle, we can determine the values of the six trigonometric functions:
1. Sine (sin):
sin(0) = Opposite/Hypotenuse = (-3)/5 = -3/5
2. Cosine (cos):
cos(0) = Adjacent/Hypotenuse = 4/5
3. Tangent (tan):
tan(0) = Opposite/Adjacent = (-3)/4 = -3/4
4. Cosecant (csc):
csc(0) = 1/sin(0) = 1/(-3/5) = -5/3
5. Secant (sec):
sec(0) = 1/cos(0) = 1/(4/5) = 5/4
6. Cotangent (cot):
cot(0) = 1/tan(0) = 1/(-3/4) = -4/3
Therefore, the values of the six trigonometric functions of an angle 0 in standard position whose terminal side passes through point P(4, -3) are:
sin(0) = -3/5
cos(0) = 4/5
tan(0) = -3/4
csc(0) = -5/3
sec(0) = 5/4
cot(0) = -4/3