solve the point given below is on the terminal side of an angle0. find the exact value of the six trigonometric functions of 0. (10,-10)
the other long side is 10sqrt2
sinTheta=-10/10sqrt2=-.707
tantheta=-10/10=-1
you do the rest, if necessary, I can check them.
To find the exact values of the six trigonometric functions of an angle, you first need to determine the values of the sides of the triangle formed by the angle.
Since the point (10, -10) is given to be on the terminal side of the angle, we can visualize a right triangle with the hypotenuse as the distance between the origin (0, 0) and the given point (10, -10).
To find the length of the sides, we can use the Pythagorean theorem:
c^2 = a^2 + b^2
In our case, the hypotenuse c is the distance between the origin and the point (10, -10), which we can find using the distance formula:
c = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(10 - 0)^2 + (-10 - 0)^2]
= √(10^2 + (-10)^2)
= √(100 + 100)
= √200
= 10√2
Now that we have the hypotenuse, we can determine the lengths of the other two sides of the triangle. We know that the x-coordinate is the adjacent side and the y-coordinate is the opposite side.
Adjacent side (a) = 10
Opposite side (b) = -10
Next, we can calculate the values of the six trigonometric functions:
1. Sine (sin θ) = opposite/hypotenuse = b/c = -10/(10√2) = -√2/2
2. Cosine (cos θ) = adjacent/hypotenuse = a/c = 10/(10√2) = √2/2
3. Tangent (tan θ) = opposite/adjacent = b/a = -10/10 = -1
4. Cosecant (csc θ) = 1/sin θ = 1/(-√2/2) = -2/√2 = -√2
5. Secant (sec θ) = 1/cos θ = 1/(√2/2) = 2/√2 = √2
6. Cotangent (cot θ) = 1/tan θ = 1/(-1) = -1
Therefore, the exact values of the trigonometric functions of the angle 0 in this case are:
sin θ = -√2/2
cos θ = √2/2
tan θ = -1
csc θ = -√2
sec θ = √2
cot θ = -1