Find the values of the six trigonometric functions of an angle in standard position if the point with coordinates (12, 5) lies on its terminal side?
Please explain how!
if the hypotenuse is at (12,5), the hypotenuse is length 13:
sinθ = 5/13
cosθ = 12/13
tanθ = 5/12
secθ = 1/cosθ = 13/12
cscθ = 1/sinθ = 13/5
cotθ = 1/tanθ = 12/5
To find the values of the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) of an angle in standard position, we can use the given point with coordinates (12, 5) on its terminal side.
1. First, we need to find the length of the hypotenuse. The hypotenuse is the distance between the origin (0, 0) and the point (12, 5). We can use the Pythagorean theorem to find its length:
hypotenuse = sqrt((12^2) + (5^2)) = sqrt(144 + 25) = sqrt(169) = 13
2. The sine of the angle is given by the ratio of the opposite side to the hypotenuse:
sine = opposite / hypotenuse = 5 / 13
3. The cosine of the angle is given by the ratio of the adjacent side to the hypotenuse:
cosine = adjacent / hypotenuse = 12 / 13
4. The tangent of the angle is given by the ratio of the opposite side to the adjacent side:
tangent = opposite / adjacent = 5 / 12
5. The cosecant of the angle is the reciprocal of the sine:
cosecant = 1 / sine = 1 / (5 / 13) = 13 / 5
6. The secant of the angle is the reciprocal of the cosine:
secant = 1 / cosine = 1 / (12 / 13) = 13 / 12
7. The cotangent of the angle is the reciprocal of the tangent:
cotangent = 1 / tangent = 1 / (5 / 12) = 12 / 5
So, the values of the six trigonometric functions for this angle are:
sine = 5 / 13
cosine = 12 / 13
tangent = 5 / 12
cosecant = 13 / 5
secant = 13 / 12
cotangent = 12 / 5
To find the values of the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) of an angle in standard position, given that a point lies on its terminal side, we can follow these steps:
1. Determine the length of the hypotenuse:
To find the length of the hypotenuse, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. Thus, we have:
hypotenuse^2 = x-coordinate^2 + y-coordinate^2
hypotenuse^2 = 12^2 + 5^2
hypotenuse^2 = 144 + 25
hypotenuse^2 = 169
hypotenuse = √169
hypotenuse = 13
2. Determine the angle:
Since the point (12, 5) lies on the terminal side of the angle, we can treat the x-coordinate as the adjacent side and the y-coordinate as the opposite side of a right triangle. Therefore, we can use the tangent formula to find the angle:
tan(angle) = opposite/adjacent
tan(angle) = 5/12
angle = tan^(-1)(5/12)
Using a calculator, angle ≈ 22.62°
3. Determine the values of the trigonometric functions:
Now that we have the angle and the length of the hypotenuse, we can use trigonometric ratios to find the values of the six trigonometric functions:
- Sine (sin): sin(angle) = opposite/hypotenuse
sin(angle) = 5/13
- Cosine (cos): cos(angle) = adjacent/hypotenuse
cos(angle) = 12/13
- Tangent (tan): tan(angle) = opposite/adjacent
tan(angle) = 5/12
- Cosecant (csc): csc(angle) = 1/sin(angle)
csc(angle) = 1/(5/13) = 13/5
- Secant (sec): sec(angle) = 1/cos(angle)
sec(angle) = 1/(12/13) = 13/12
- Cotangent (cot): cot(angle) = 1/tan(angle)
cot(angle) = 1/(5/12) = 12/5
Therefore, the values of the six trigonometric functions for the angle whose terminal side has the point (12, 5) are:
sin(angle) = 5/13,
cos(angle) = 12/13,
tan(angle) = 5/12,
csc(angle) = 13/5,
sec(angle) = 13/12,
cot(angle) = 12/5.