If the terminal side of an angle theta contains the point (-1,3), find secant theta.
You should memorize the trig ratios is terms of x, y and r
so for the point (-1,3)
x = -1, y = 3 and by Pythagoras, r = √10
cosθ = x/r = -1/√10
and since secθ = 1/cosθ
secθ = -√10
To find the secant of an angle, we need to find the length of the hypotenuse. The hypotenuse is the distance between the origin (0,0) and the given point (-1,3).
We can use the Pythagorean theorem to find the length of the hypotenuse. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the two other sides (a and b). In this case, one side will be the x-coordinate (-1) and the other side will be the y-coordinate (3).
Let's calculate the hypotenuse:
c² = a² + b²
c² = (-1)² + 3²
c² = 1 + 9
c² = 10
To find c, we take the square root of both sides:
c = sqrt(10)
The secant of an angle is defined as the reciprocal of the cosine of the angle. The cosine of an angle is equal to the ratio of the adjacent side to the hypotenuse in a right triangle. In this case, the hypotenuse is sqrt(10) and the adjacent side is -1.
Therefore, the secant of theta is equal to the reciprocal of the cosine of theta:
sec(theta) = 1 / cos(theta)
cos(theta) = adjacent / hypotenuse
cos(theta) = -1 / sqrt(10)
So, the secant of theta is:
sec(theta) = 1 / (-1 / sqrt(10))
sec(theta) = -sqrt(10)