P(-5, 5) is a point on the terminal side of θ in standard position. What is the exact value of sec θ?
your are in quad II
drop a perpendicular to the x-axis and draw your right-angled triangle
cosØ = x/r
secØ = r/x
remember : r^2 = x^2 + y^2
= (-5)^2 + 5^2
= 50
r = 5√2
secØ = 5√2/-5 = -√2
Well, I hope you're ready for a bad joke, because here it comes:
Why did the angle bring a ladder to the party?
Because it wanted to reach the sec-theta!
Ah, I crack myself up. In all seriousness, though, let's solve this.
We know that the point (-5, 5) is on the terminal side of the angle θ in standard position. To find the exact value of sec θ, we can use the Pythagorean Theorem.
From the point (-5, 5), we can form a right triangle by drawing a perpendicular line to the x-axis. The base of the triangle will be 5 units, and the height will also be 5 units.
Now, let's calculate the length of the hypotenuse using the Pythagorean Theorem:
c^2 = a^2 + b^2
c^2 = 5^2 + 5^2
c^2 = 25 + 25
c^2 = 50
c = √50
So, the length of the hypotenuse is √50.
Now, sec θ is the reciprocal of the cosine of θ. The cosine of an angle is equal to the adjacent side over the hypotenuse. In this case, the adjacent side is -5 and the hypotenuse is √50.
So sec θ = 1/cos θ = 1/(-5/√50) = -√50/5.
Therefore, the exact value of sec θ is -√50/5. Just remember to keep a calculator handy in case you need to simplify further!
To find the exact value of sec θ (secant of θ), we need to determine the cosine of θ first.
Given that the point P(-5, 5) is on the terminal side of θ, we can use the coordinates to determine the trigonometric ratios.
Let's find the length of the hypotenuse (r) using the Pythagorean theorem:
r^2 = (-5)^2 + 5^2
r^2 = 25 + 25
r^2 = 50
r = √50
r = 5√2
Now, we can determine the cosine of θ using the x-coordinate of the point P:
cos θ = adjacent side / hypotenuse = x / r = -5 / (5√2) = -√2 / 2
Finally, we can calculate the secant of θ, which is the reciprocal of the cosine:
sec θ = 1 / cos θ = 1 / (-√2 / 2) = -2 / √2 = (-2 / √2) * (√2 / √2) = -2√2 / 2 = -√2
To find the exact value of sec θ, we need to use the coordinates of the point (-5, 5) on the terminal side of angle θ in standard position.
In standard position, a point on the terminal side of an angle is determined by the coordinates (x, y), where x and y represent the horizontal and vertical distances from the origin (0, 0), respectively.
Since the given point is (-5, 5), we can determine the values of trigonometric ratios using these coordinates.
First, we need to find the hypotenuse of the right triangle formed by the given point and the origin. To do this, we can use the Pythagorean theorem.
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 lengths of the other two sides (a and b). In this case, the hypotenuse will be the distance between the origin and the given point.
Using the coordinates (-5, 5), we can calculate the length of the hypotenuse (c) as follows:
c = sqrt((-5)^2 + 5^2)
= sqrt(25 + 25)
= sqrt(50)
= 5 sqrt(2)
Now, we can find the exact value of sec θ. The secant of an angle is defined as the reciprocal of the cosine of the angle.
In the given triangle, the adjacent side is -5 and the hypotenuse is 5 sqrt(2). Therefore, the cosine of θ can be calculated as:
cos θ = adjacent/hypotenuse
= -5/(5 sqrt(2))
= -1/sqrt(2)
Finally, taking the reciprocal of the cosine, we can find the exact value of sec θ:
sec θ = 1/cos θ
= 1/(-1/sqrt(2))
= -sqrt(2)
Therefore, the exact value of sec θ is -sqrt(2).