Determine the exact values of the six trigonometric functions of the angle.(-5,-2)
You have x,y
r^2 = x^2+y^2, so r = √29
Now just plug and chug:
sin = y/r
cos = x/r
tan = y/x
and so on
To determine the exact values of the six trigonometric functions of an angle, you need to find the values of the sine, cosine, tangent, cosecant, secant, and cotangent. In order to find the angle, we can use the inverse trigonometric functions.
Let's start by finding the angle:
1. Use the point (-5, -2) to determine the length of the hypotenuse and the other two sides of the right triangle. Since the x-coordinate is -5 and the y-coordinate is -2, we can label the adjacent side as -5 and the opposite side as -2.
2. Use the Pythagorean theorem to find the hypotenuse:
hypotenuse^2 = adjacent^2 + opposite^2
hypotenuse^2 = (-5)^2 + (-2)^2
hypotenuse^2 = 25 + 4
hypotenuse^2 = 29
hypotenuse = √29
3. Use the definitions of sine and cosine to find the angle:
sine(angle) = opposite / hypotenuse
sine(angle) = -2 / √29
cosine(angle) = adjacent / hypotenuse
cosine(angle) = -5 / √29
4. Take the inverse of sine and cosine to find the angle:
angle = arcsin(-2 / √29)
angle = arccos(-5 / √29)
Now that we have the value of the angle, we can find the values of the six trigonometric functions:
Sine: sin(angle) = -2 / √29
Cosine: cos(angle) = -5 / √29
Tangent: tan(angle) = sin(angle) / cos(angle)
tan(angle) = (-2 / √29) / (-5 / √29) = 2 / 5
Cosecant: csc(angle) = 1 / sin(angle)
csc(angle) = 1 / (-2 / √29) = -√29 / 2
Secant: sec(angle) = 1 / cos(angle)
sec(angle) = 1 / (-5 / √29) = -√29 / 5
Cotangent: cot(angle) = 1 / tan(angle)
cot(angle) = 1 / (2 / 5) = 5 / 2
Therefore, the exact values for the six trigonometric functions of the given angle are:
sine = -2 / √29
cosine = -5 / √29
tangent = 2 / 5
cosecant = -√29 / 2
secant = -√29 / 5
cotangent = 5 / 2