The point (-3. -6) lies on the terminal arm of an angle Θ in standard position.
a) Determine the exact value of sin Θ, cosΘ, tanΘ
b) Determine the principal angle and the related acute angle.
To solve this problem, we can use the Pythagorean Theorem and the definitions of the trigonometric functions.
a) First, let's find the magnitude of the vector from the origin to the point (-3, -6). Since the x-coordinate is -3 and the y-coordinate is -6, we have:
Magnitude = √((-3)^2 + (-6)^2) = √(9 + 36) = √45
Now, let's find the values of sin Θ, cos Θ, and tan Θ:
sin Θ = y / Magnitude = -6 / √45
cos Θ = x / Magnitude = -3 / √45
tan Θ = sin Θ / cos Θ = (-6 / √45) / (-3 / √45) = 2
b) To find the principal angle, we can use the inverse tangent function:
Principal Angle = arctan(tan Θ) = arctan(2)
To find the related acute angle, we can take the absolute value of the principal angle:
Related Acute Angle = |Principal Angle| = |arctan(2)|