If θ is an angle in standard position and its terminal side passes through the point (20,-21), find the exact value of sec(θ).
its 21/29
If θ is an angle in standard position and its terminal side passes through the point (20,21), find the exact value of \cos\thetacosθ in simplest radical form.
its 2/9
Why did the angle go to the therapist? Because it was feeling a bit terminal! But no worries, let's find the exact value of sec(θ) for you.
First, let's find the hypotenuse of the right triangle formed by the point (20, -21). Using the Pythagorean theorem, we have:
hypotenuse^2 = 20^2 + (-21)^2
hypotenuse^2 = 400 + 441
hypotenuse^2 = 841
Taking the square root of both sides, we get:
hypotenuse = √841
hypotenuse = 29
Since sec(θ) is equal to the hypotenuse divided by the adjacent side, we can conclude that sec(θ) = 29/20.
An exact value, without any decimal approximation! Hope that sheds some funny light on sec(θ) for you!
To find the exact value of sec(θ), we need to first determine the position of the angle θ in standard position.
Since the terminal side of θ passes through the point (20, -21), we can determine the values of the adjacent side and the hypotenuse.
Let's denote the length of the adjacent side as x and the length of the hypotenuse as r. We can then use the Pythagorean theorem to find the value of x.
The Pythagorean theorem states that:
r^2 = x^2 + y^2
In this case, we have:
r^2 = x^2 + (-21)^2
r^2 = x^2 + 441
Now, let's consider the definition of sec(θ):
sec(θ) = r / x
Substituting the expression for r^2 from above, we have:
sec(θ) = √(x^2 + 441) / x
Now, to find the exact value of sec(θ), we need to determine the value of x. One way to do this is by using trigonometric ratios. Since we know the coordinates of the point (20, -21), we can calculate the value of x using the inverse tangent function.
Let's use the following trigonometric ratios:
tan(θ) = y / x
Substituting the values, we have:
tan(θ) = (-21) / 20
Taking the inverse tangent of both sides, we get:
θ = arctan((-21) / 20)
Using a calculator, compute the value of arctan((-21) / 20). The result is approximately -48.37 degrees.
With this angle, we can now determine the exact value of x using the trigonometric ratios:
tan(θ) = (-21) / 20
x = 20 / cos(θ)
Substituting the value of θ, we have:
x = 20 / cos(-48.37)
Using a calculator, compute the value of cos(-48.37). The result is approximately 0.669.
Now, substitute the value of x into the expression for sec(θ):
sec(θ) = √(x^2 + 441) / x
sec(θ) = √((20 / cos(-48.37))^2 + 441) / (20 / cos(-48.37))
Calculate this expression to find the exact value of sec(θ).
so x = 20 , y = -21 , which is in quad IV
r^2= 400+441 = 841
r = √841 = 29
then cos Ø = 20/29
sec Ø = 29/20 = 1.45