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!

plot the point (12,5) and join it to the origin.

draw a vertical line from the point to the x-axis
You now have a right-angled triangle with a base (x) of 12 and a height (y) of 5
Using Pythagoras, we can find the hypotenuse
h^2 = 12^2 + 5^2 = 169
h = 13
let the base angle be Ø

You should have learned the following definitions
sinØ = opposite/hypotenuse = 5/13
cosØ = adjacent/hypotenus = 12/13
tanØ = opposite/adjacent = 5/12

of course ...
cscØ = 13/5
secØ = 13/12
cotØ = 12/5

I know the answer but I don't understand the process. ! If you could explain the process without the answers?!

To find the values of the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) of an angle in standard position, given that the point (12, 5) lies on its terminal side, follow these steps:

1. Determine the length of the hypotenuse, r:
- The distance from the origin (0,0) to the point (12, 5) is the hypotenuse, which can be found using the Pythagorean theorem:
- r² = x² + y²
- r² = 12² + 5²
- r² = 144 + 25
- r² = 169
- r = √169
- r = 13

2. Determine the values of the trigonometric functions:
- Sine (sin): sinθ = y/r = 5/13 ≈ 0.385
- Cosine (cos): cosθ = x/r = 12/13 ≈ 0.923
- Tangent (tan): tanθ = y/x = 5/12 ≈ 0.417
- Cosecant (csc): cscθ = 1/sinθ = 1/0.385 ≈ 2.597
- Secant (sec): secθ = 1/cosθ = 1/0.923 ≈ 1.083
- Cotangent (cot): cotθ = 1/tanθ = 1/0.417 ≈ 2.401

So, the values of the six trigonometric functions of the angle in standard position are approximately:
sinθ ≈ 0.385
cosθ ≈ 0.923
tanθ ≈ 0.417
cscθ ≈ 2.597
secθ ≈ 1.083
cotθ ≈ 2.401

To find the values of the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) of an angle in standard position when a point with coordinates (x, y) lies on its terminal side, you can use the following steps:

1. Determine the distance from the origin to the point using the Pythagorean theorem: d = √(x^2 + y^2). In this case, the distance is d = √(12^2 + 5^2) = √(144 + 25) = √169 = 13.

2. Determine the angle θ in standard position by using the formula: tan(θ) = y / x, where θ is the angle in standard position, y is the y-coordinate of the point, and x is the x-coordinate of the point. In this case, tan(θ) = 5 / 12. Taking the inverse tangent (arctan) of both sides gives us θ = arctan(5 / 12).

3. Find the values of the six trigonometric functions using the angle θ:

- Sine (sin): sin(θ) = y / d = 5 / 13.
- Cosine (cos): cos(θ) = x / d = 12 / 13.
- Tangent (tan): tan(θ) = y / x = 5 / 12.
- Cosecant (csc): csc(θ) = 1 / sin(θ) = 13 / 5.
- Secant (sec): sec(θ) = 1 / cos(θ) = 13 / 12.
- Cotangent (cot): cot(θ) = 1 / tan(θ) = 12 / 5.

Therefore, the values of the six trigonometric functions of the angle in standard position are:
sin(θ) = 5 / 13,
cos(θ) = 12 / 13,
tan(θ) = 5 / 12,
csc(θ) = 13 / 5,
sec(θ) = 13 / 12,
cot(θ) = 12 / 5.