Let (7,-3) be a point on the terminal side of theta. Find the exact values of sin of theta, csc of theta and cot of theta?
To find the exact values of sin(theta), csc(theta), and cot(theta), we need to consider the point (7, -3) on the unit circle.
Step 1: Determine the radius of the unit circle by using the Pythagorean theorem. The radius can be found by calculating the square root of the sum of squares of the x and y coordinates of the point on the unit circle.
r = sqrt((7^2) + (-3^2))
r = sqrt(49 + 9)
r = sqrt(58)
Step 2: Using the values of x and y on the unit circle, we can calculate sin(theta) and csc(theta).
sin(theta) = y / r
sin(theta) = -3 / sqrt(58)
csc(theta) = 1 / sin(theta)
csc(theta) = sqrt(58) / -3
Step 3: Now we can find cot(theta) by using the values of x and y on the unit circle.
cot(theta) = x / y
cot(theta) = 7 / -3
So, the exact values of sin(theta), csc(theta), and cot(theta) are:
sin(theta) = -3 / sqrt(58)
csc(theta) = sqrt(58) / -3
cot(theta) = 7 / -3
To find the exact values of sin, csc, and cot of theta, we can use the given point (7, -3) on the terminal side of theta.
First, let's find the hypotenuse of the right triangle formed by the given point (7, -3). This can be done using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Using the coordinates (7, -3), we have:
Hypotenuse^2 = 7^2 + (-3)^2
Hypotenuse^2 = 49 + 9
Hypotenuse^2 = 58
Taking the square root of both sides, we find the hypotenuse:
Hypotenuse = √58
Now, we can calculate the sine (sin) of theta. The sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse. In this case, the side opposite theta is -3 and the hypotenuse is √58. Thus:
sin(theta) = -3 / √58
Next, we can find the cosecant (csc) of theta, which is the reciprocal of the sine. Therefore:
csc(theta) = 1 / sin(theta)
csc(theta) = 1 / (-3 / √58)
csc(theta) = -√58 / 3
Finally, we can find the cotangent (cot) of theta. The cotangent of an angle is equal to the adjacent side divided by the opposite side. In this case, the adjacent side is 7 and the opposite side is -3. Thus:
cot(theta) = 7 / -3
So, the exact values for sin(theta), csc(theta), and cot(theta) are:
sin(theta) = -3 / √58
csc(theta) = -√58 / 3
cot(theta) = -7 / 3
y = -3
x = 7
r = √(9+49) = √58
Now recall that
sinθ = y/r
cosθ = x/r
tanθ = y/x