Let µ be an angle in standard position that contains the point with co-
ordinates (-5;-12) on its terminal side. Determine the values of the six trigono-
metric functions of angle µ.
Draw the triangle. The horizontal length is -5, altitude is -12, and you do the calculating, hypotenuse is 13
Cosine Angle= -5/13 and so forth.
To determine the values of the trigonometric functions of angle µ, we can use the coordinates of the point (-5, -12) on the terminal side of the angle. The trigonometric functions are based on the ratios of the sides of a right triangle formed with the angle µ.
Step 1: Find the length of the hypotenuse of the right triangle
We can use the Pythagorean theorem to find the length of the hypotenuse. The formula is c^2 = a^2 + b^2, where c is the hypotenuse and a and b are the lengths of the other two sides. In this case, the sides are represented by the coordinates (-5, -12). Therefore, we have:
c^2 = (-5)^2 + (-12)^2
c^2 = 25 + 144
c^2 = 169
c = √169
c = 13
Step 2: Find the values of the trigonometric functions
a) sine (sin): Sin(µ) = opposite / hypotenuse
In this case, the opposite side is the y-coordinate (-12) and the hypotenuse is 13. So we have:
sin(µ) = -12 / 13
b) cosine (cos): Cos(µ) = adjacent / hypotenuse
The adjacent side is the x-coordinate (-5) and the hypotenuse is 13. So we have:
cos(µ) = -5 / 13
c) tangent (tan): Tan(µ) = opposite / adjacent
The opposite side is -12 and the adjacent side is -5. So we have:
tan(µ) = -12 / -5
d) cosecant (csc): Csc(µ) = 1 / sin(µ)
Recall that sin is -12/13, so we have:
csc(µ) = 1 / (-12 / 13)
e) secant (sec): Sec(µ) = 1 / cos(µ)
Recall that cos is -5/13, so we have:
sec(µ) = 1 / (-5 / 13)
f) cotangent (cot): Cot(µ) = 1 / tan(µ)
Recall that tan is -12/-5, so we have:
cot(µ) = 1 / (-12 / -5)
Now you can substitute the values of the trigonometric functions into these formulas and calculate their respective values.