Find the six trigonometric functions for an angle in standard position whose terminal ray passes through (5, -2).
finish the triangle. You have
x = 5
y = -2
r = √29
Now recall that
sinθ = y/r
cosθ = x/r
tanθ = y/x
and so on.
To find the six trigonometric functions for an angle in standard position, we first need to determine the values of the sides of the right triangle formed by the angle.
Given that the terminal ray passes through (5, -2), we can use the distance formula to find the length of the hypotenuse, which is the distance from the origin to (5, -2):
h = √((5 - 0)^2 + (-2 - 0)^2)
= √(25 + 4)
= √29
Next, we can find the values of the adjacent and opposite sides of the triangle. The adjacent side is the x-coordinate of the point (5, -2) and the opposite side is the y-coordinate. Therefore:
adjacent = 5
opposite = -2
Now, we can use these side lengths to determine the six trigonometric functions:
1. The sine function (sin) is defined as the ratio of the opposite side to the hypotenuse:
sin = opposite / hypotenuse
= -2 / √29
= -2√29 / 29 (rationalizing the denominator)
2. The cosine function (cos) is defined as the ratio of the adjacent side to the hypotenuse:
cos = adjacent / hypotenuse
= 5 / √29
= 5√29 / 29 (rationalizing the denominator)
3. The tangent function (tan) is defined as the ratio of the opposite side to the adjacent side:
tan = opposite / adjacent
= -2 / 5
4. The cosecant function (csc) is defined as the reciprocal of the sine function:
csc = 1 / sin
= 29 / -2√29
= -29√29 / 2 (rationalizing the denominator)
5. The secant function (sec) is defined as the reciprocal of the cosine function:
sec = 1 / cos
= 29 / 5√29
= 29√29 / 5 (rationalizing the denominator)
6. The cotangent function (cot) is defined as the reciprocal of the tangent function:
cot = 1 / tan
= 5 / -2
= -5/2
So, the six trigonometric functions for an angle in standard position whose terminal ray passes through (5, -2) are:
sin = -2√29 / 29
cos = 5√29 / 29
tan = -2 / 5
csc = -29√29 / 2
sec = 29√29 / 5
cot = -5/2
To find the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) for an angle in standard position whose terminal ray passes through the point (5, -2), we can use the given coordinates to determine the values of the sides of a right triangle.
Let's label the sides of the triangle:
- hypotenuse (h) = the distance from the origin to the point (5, -2)
- adjacent side (a) = the x-coordinate of the point (5, -2)
- opposite side (o) = the y-coordinate of the point (5, -2)
Using the Pythagorean theorem, we can calculate the length of the hypotenuse:
h = sqrt(a^2 + o^2)
Plugging in the values, we get:
h = sqrt(5^2 + (-2)^2) = sqrt(25 + 4) = sqrt(29)
Now, we can calculate the trigonometric functions:
1. Sine (sinθ) = o/h
sinθ = (-2) / sqrt(29)
2. Cosine (cosθ) = a/h
cosθ = 5 / sqrt(29)
3. Tangent (tanθ) = o/a
tanθ = (-2) / 5
4. Cosecant (cscθ) = 1/sinθ
cscθ = 1 / sinθ
5. Secant (secθ) = 1/cosθ
secθ = 1 / cosθ
6. Cotangent (cotθ) = 1/tanθ
cotθ = 1 / tanθ
Therefore, the six trigonometric functions for the angle in standard position whose terminal ray passes through (5, -2) are:
- Sine (sinθ) = (-2) / sqrt(29)
- Cosine (cosθ) = 5 / sqrt(29)
- Tangent (tanθ) = (-2) / 5
- Cosecant (cscθ) = 1 / sinθ
- Secant (secθ) = 1 / cosθ
- Cotangent (cotθ) = 1 / tanθ