If θ is an angle in standard position and its terminal side passes through the point (-3,-7), find the exact value of
sec
θ
secθ in simplest radical form.
r^2 = 3^2 + 7^2
secθ = r/x
To find the exact value of secθ in simplest radical form, we need to use the given information that θ is an angle in standard position and its terminal side passes through (-3, -7).
First, let's find the length of the hypotenuse by using the Pythagorean theorem:
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Let's denote the horizontal side as the adjacent side (a) and the vertical side as the opposite side (b). In this case, a = -3 and b = -7.
Using the Pythagorean theorem, we can calculate the length of the hypotenuse (c):
c^2 = a^2 + b^2
c^2 = (-3)^2 + (-7)^2
c^2 = 9 + 49
c^2 = 58
Taking the square root of both sides, we get:
c = √58
Now, we know that the secant function is defined as the reciprocal of the cosine function:
secθ = 1 / cosθ
To find cosθ, we can use the triangle we formed earlier. Since the adjacent side is -3 and the hypotenuse is √58, we can find cosθ as the ratio of the adjacent side to the hypotenuse:
cosθ = a / c
cosθ = -3 / √58
To simplify this expression, we multiply the numerator and denominator by √58:
cosθ = (-3 / √58) * (√58 / √58)
cosθ = -3√58 / 58
Now, we can find secθ by taking the reciprocal of cosθ:
secθ = 1 / cosθ
secθ = 1 / (-3√58 / 58)
secθ = 58 / (-3√58)
To simplify this expression, we can simplify the fraction by dividing the numerator and denominator by the greatest common divisor, which is 1:
secθ = -58 / (3√58)
Therefore, the exact value of secθ in simplest radical form is -58 / (3√58).