P is a point on the terminal side of θ in standard position. Find the exact value of ALL SIX trigonometric functions.
P (5, −12)
To find the exact value of all six trigonometric functions for the point P(5, -12) on the terminal side of angle θ in standard position, we can use the coordinates of the point (x, y) and the distance formula.
First, let's calculate the distance from the origin to point P using the Pythagorean theorem:
d = √(x^2 + y^2)
= √(5^2 + (-12)^2)
= √(25 + 144)
= √169
= 13
Now, we can use the given coordinates (x, y) to find the values of the trigonometric functions.
1. Sine (sinθ):
Sinθ = y / d
= -12 / 13
2. Cosine (cosθ):
Cosθ = x / d
= 5 / 13
3. Tangent (tanθ):
Tanθ = y / x
= -12 / 5
4. Cosecant (cscθ):
Cscθ = 1 / sinθ
= 1 / (-12 / 13)
= -13 / 12
5. Secant (secθ):
Secθ = 1 / cosθ
= 1 / (5 / 13)
= 13 / 5
6. Cotangent (cotθ):
Cotθ = 1 / tanθ
= 1 / (-12 / 5)
= -5 / 12
Therefore, the exact values of all six trigonometric functions for the point P(5, -12) are:
sinθ = -12 / 13
cosθ = 5 / 13
tanθ = -12 / 5
cscθ = -13 / 12
secθ = 13 / 5
cotθ = -5 / 12