The terminal side of an angle alpha (in standard position) contains the point (-3,-4). Find sec alpha.
did you recognize the 3-4-5 rightangled triangle ?
cosA = -3/5
so secA -5/3
Thank you for that! I forgot how to start it!
To find the secant of angle alpha, we need to determine the value of the hypotenuse of a right triangle formed by the point (-3,-4).
Let's label the sides of the triangle as follows:
- The hypotenuse is labeled as `h`.
- The side adjacent to angle alpha is labeled as `x`.
- The side opposite to angle alpha is labeled as `y`.
We can use the Pythagorean theorem to find the value of `h`, which is the length of the hypotenuse:
h^2 = x^2 + y^2
Substituting the given values of x=-3 and y=-4:
h^2 = (-3)^2 + (-4)^2
= 9 + 16
= 25
Taking the square root of both sides, we get:
h = √25
= 5
Therefore, the length of the hypotenuse `h` is 5.
Now, the secant of angle alpha is defined as the reciprocal of the cosine of the angle:
sec alpha = 1 / cos alpha
To find the cosine of angle alpha, we can use the formula:
cos alpha = x / h
Substituting the values of x=-3 and h=5:
cos alpha = -3 / 5
Finally, taking the reciprocal of cos alpha, we have:
sec alpha = 1 / ( -3 / 5 )
= 5 / -3
So, the secant of angle alpha is -5/3.
To find sec alpha, we need to use the given point (-3, -4) on the terminal side of the angle in standard position.
First, we can use the coordinates of the point (-3, -4) to find the value of the hypotenuse in a right triangle. The hypotenuse is the distance from the origin (0, 0) to the point (-3, -4).
Using the distance formula, we have:
Distance = √[(x2 - x1)² + (y2 - y1)²]
Distance = √[(-3 - 0)² + (-4 - 0)²]
Distance = √[9 + 16]
Distance = √25
Distance = 5
So, the length of the hypotenuse is 5.
Next, we need to find the value of sec alpha, which is the reciprocal of the cosine of alpha.
Since the point (-3, -4) is in the third quadrant, we can determine the values of the adjacent side and the hypotenuse in the right triangle formed.
In the third quadrant, the adjacent side is negative and the hypotenuse is positive.
Using the Pythagorean theorem, we can find the length of the opposite side:
Opposite Side = √(Hypotenuse² - Adjacent Side²)
Opposite Side = √(5² - (-3)²)
Opposite Side = √(25 - 9)
Opposite Side = √16
Opposite Side = 4
Now, we have the lengths of the adjacent side and the hypotenuse. Using these values, we can find the cosine of alpha:
cos(alpha) = Adjacent Side / Hypotenuse
cos(alpha) = -3 / 5
Finally, to find sec(alpha), we take the reciprocal of the cosine:
sec(alpha) = 1 / cos(alpha)
sec(alpha) = 1 / (-3 / 5)
sec(alpha) = -5 / 3
Therefore, the value of sec alpha is -5/3.