Suppose that
θ
is an angle in standard position whose terminal side intersects the unit circle at
, −15/17,−8/17
Θ is in quad III
the tangent of the reference angle is
... 8/15
To determine the value of θ, we can use the coordinates of the point where the terminal side intersects the unit circle.
Given that the point is (-15/17, -8/17), we can use the inverse trigonometric functions to find the angle.
Let's first find the value of the hypotenuse using the Pythagorean theorem:
hypotenuse = √((-15/17)^2 + (-8/17)^2)
= √((225/289) + (64/289))
= √(289/289)
= 1
Now, let's find the values of sine and cosine using the coordinates (-15/17, -8/17):
sine (θ) = opposite/hypotenuse
= (-8/17) / 1
= -8/17
cosine (θ) = adjacent/hypotenuse
= (-15/17) / 1
= -15/17
To find the angle θ in standard position, we can now use the inverse trigonometric functions:
θ = inverse sine (-8/17)
= -0.4811 rad (approximately)
Alternatively, we can also use the inverse cosine function:
θ = inverse cosine (-15/17)
= 2.1360 rad (approximately)
Therefore, θ is approximately -0.4811 rad or 2.1360 rad in standard position.
To find the value of θ, we can use the coordinates of the point where the terminal side intersects the unit circle.
Let's use the Pythagorean theorem to find the length of the hypotenuse of the triangle formed by the x-coordinate and y-coordinate of the point:
hypotenuse = √((x-coordinate)^2 + (y-coordinate)^2)
= √((-15/17)^2 + (-8/17)^2)
= √(225/289 + 64/289)
= √(289/289)
= 1
Since we are dealing with a unit circle (which has a radius of 1), the length of the hypotenuse is 1.
Now, to find θ, we need to find the angle formed by the terminal side of the triangle and the positive x-axis. We can use trigonometry for this.
sin(θ) = (opposite side)/(hypotenuse)
= (-8/17) / 1
= -8/17
Let's find the angle θ using the inverse sine function (sin^(-1)):
θ = sin^(-1)(-8/17)
≈ -0.4966 radians or -28.49 degrees
So, the angle θ is approximately -0.4966 radians or -28.49 degrees.