Find the exact values of tangent and cosecant for angle theta in standard position whose terminal side passes through the point (16,-4)
the angle is in QIV.
tanθ = -4/16, so
cscθ = -4/4√17 = -1/√17
find the exact value of sec theta for angle theta in standard position if the point -3,2 lies on its terminal side
To find the exact values of tangent and cosecant for angle theta, we first need to determine the coordinates of the point at which the terminal side of theta intersects the unit circle in standard position.
Given that the terminal side of theta passes through the point (16, -4), we can find the length of the hypotenuse of the right triangle formed by this point on the coordinate plane.
Using the Pythagorean theorem, we know that:
hypotenuse^2 = (16^2) + (-4^2)
hypotenuse^2 = 256 + 16
hypotenuse^2 = 272
Taking the square root of both sides, we find:
hypotenuse = √272
hypotenuse = 16.49 (rounded to two decimal places)
Next, we need to determine the values of the adjacent side and opposite side of the triangle. Since the given point falls in the second quadrant (quadrant II), the x-coordinate (16) is positive, and the y-coordinate (-4) is negative.
The adjacent side of the triangle is the x-coordinate, so adjacent = 16.
And the opposite side of the triangle is the absolute value of the y-coordinate, so opposite = |-4| = 4.
Now, we can calculate the exact values of tangent and cosecant.
Tangent (θ) = opposite / adjacent
Tangent (θ) = 4 / 16
Tangent (θ) = 1 / 4
Tangent (θ) = 0.25
Cosecant (θ) = 1 / sine (θ)
To find sine (θ), we can use the formula:
sine (θ) = opposite / hypotenuse
sine (θ) = 4 / 16.49
sine (θ) = 0.2424 (rounded to four decimal places)
Cosecant (θ) = 1 / sine (θ)
Cosecant (θ) = 1 / 0.2424
Cosecant (θ) = 4.1252 (rounded to four decimal places)
Therefore, the exact values of tangent (θ) and cosecant (θ) for the angle theta are:
Tangent (θ) = 1/4 or 0.25
Cosecant (θ) = 4.1252