Find the 6 trigonometric function of teta if it's terminal side passes through the point(1,2)

so you know that x (adjacent) is 1 and

y (opposite) is 2

Using Pythagoras we can find r (the hypotenuse)
r^2 = 1^2 + 2^2 = 5
r = √5

now now simply use the definitions to form the 6 different trig ratios
e.g.
cos Ø = adj/hyp = 1/√5
sec Ø = 1/cosØ = √5
etc

To find the trigonometric functions of an angle, we need to know the position of the angle in the coordinate plane, specifically the point where the terminal side passes through.

In this case, the terminal side of the angle passes through the point (1,2). Let's label this point as P.

To find the trigonometric functions, we need to determine the values of the three sides of the right triangle formed by the terminal side of the angle and the x and y-axes.

Since point P is on the terminal side of the angle, we can determine the length of the hypotenuse, as well as the lengths of the adjacent and opposite sides using the coordinates of point P.

The hypotenuse is the distance from the origin (0, 0) to point P, which can be calculated using the distance formula:

hypotenuse = sqrt(x^2 + y^2)

For point P (1, 2), the hypotenuse = sqrt(1^2 + 2^2) = sqrt(1 + 4) = sqrt(5).

The adjacent side is the distance from point P to the y-axis (vertically), which is 1 in this case.

The opposite side is the distance from point P to the x-axis (horizontally), which is 2 in this case.

Now that we know the lengths of the three sides of the right triangle, we can find the six trigonometric functions of the angle.

1. Sine (sin): sin(teta) = opposite/hypotenuse = 2/sqrt(5)

2. Cosine (cos): cos(teta) = adjacent/hypotenuse = 1/sqrt(5)

3. Tangent (tan): tan(teta) = opposite/adjacent = 2/1 = 2

4. Cosecant (csc): csc(teta) = 1/sin(teta) = sqrt(5)/2

5. Secant (sec): sec(teta) = 1/cos(teta) = sqrt(5)

6. Cotangent (cot): cot(teta) = 1/tan(teta) = 1/2

So, the six trigonometric functions of the angle with the terminal side passing through the point (1, 2) are:
sin(teta) = 2/sqrt(5)
cos(teta) = 1/sqrt(5)
tan(teta) = 2
csc(teta) = sqrt(5)/2
sec(teta) = sqrt(5)
cot(teta) = 1/2