Let (-6,-5) be a point on the terminal side of theta find the exact value of sin theta, csc theta, and cot theta

Part 1)  or

Part 2)

Part 3)

Step-by-step explanation:

we know that

The point (-6,-5) lies on the III Quadrant

so

sin(theta) is negative

csc(theta) is negative

cot(theta) is positive

Part 1) Find the sine of angle theta

we know that

The function sine is equal to divide the opposite side to the angle theta by the hypotenuse

we have

Applying the Pythagoras Theorem

substitute

-----> the hypotenuse

substitute

Simplify

Remember that the sin(theta) is negative

so

Part 2) Find the cosecant of angle theta

we know that

The function cosecant is equal to divide the hypotenuse by the opposite  side to the angle theta

we have

so

Part 3) Find the cotangent of angle theta

we know that

The function cotangent is equal to divide the adjacent side to the angle theta by the opposite side to the angle theta

we have

substitute

(sin 42)/31. = sinx)/37

sinx = 37 (sin42)/31

= 37× 0.66/31 = 0.787

x= 51.9

other angle opposite to y = 180-51.9-42

= 180- 93.9 =86.1

so

sin42)/31 = sin 86.1)/y

0.021 = 0.997 /y

y= 0.997/0.021 =47.47