If θ is an angle in standard position and its
terminal side passes through the point (−3,2), find
the exact value of cscθ .
I know the answer...but i want to know how you get the answer...
The answer is
radical 13/2
If θ is an angle in standard position and its terminal side passes through the point (4,1), find the exact value of \cot\thetacotθ in simplest radical form.
Well, well, well! It seems like you're in the mood for some trigonometry fun!
To find the exact value of cscθ, we're going to have to do some digging. First, we need to find the hypotenuse of the right triangle formed by the terminal side passing through (-3, 2). We can use the Pythagorean theorem for that.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). In this case, let's call the horizontal side a and the vertical side b.
So, a = -3 and b = 2. Applying the Pythagorean theorem, we have:
c^2 = (-3)^2 + 2^2
c^2 = 9 + 4
c^2 = 13
Now, to find the exact value of cscθ, we need to remember that cscθ is the reciprocal of sinθ. And sinθ is defined as the ratio of the opposite side (b) to the hypotenuse (c). So,
cscθ = 1 / sinθ
cscθ = 1 / (b/c)
cscθ = c/b
cscθ = c / 2
Since c^2 = 13 from our previous calculations, we can substitute that value in:
cscθ = √13 / 2
And voila! The exact value of cscθ is √13 / 2.
I hope that helped, and if you have any more trigonometry questions, I'll be here all night!
To find the exact value of cscθ, we can use the Pythagorean theorem.
In the coordinate system, the point (-3,2) is located in the second quadrant.
The hypotenuse of the right triangle formed by the terminal side passing through (-3,2) can be found using the distance formula:
d = sqrt((-3)^2 + 2^2)
= sqrt(9 + 4)
= sqrt(13)
Now, cscθ is the reciprocal of the sine function.
The sine function is defined as the opposite side divided by the hypotenuse:
sinθ = opposite / hypotenuse
In this case, the opposite side is the y-coordinate, which is 2.
So, sinθ = 2 / sqrt(13)
To find cscθ, we take the reciprocal of sinθ:
cscθ = 1 / sinθ
= 1 / (2 / sqrt(13))
To rationalize the denominator, we multiply the numerator and denominator by sqrt(13):
cscθ = (1 * sqrt(13)) / (2)
= sqrt(13) / 2
Therefore, the exact value of cscθ is sqrt(13) / 2 or radical 13/2.
To find the exact value of cscθ, we need to determine the length of the hypotenuse and the length of the opposite side with respect to angle θ.
First, let's plot the given point (-3,2) on the coordinate plane. This point represents the terminal side of angle θ in standard position.
Next, we can form a right triangle by drawing a line from the point (-3,2) perpendicular to the x-axis, intersecting it at a point (let's call it A).
Now, we have a right triangle with side lengths as follows:
- The hypotenuse is the distance between the origin (0,0) and point A.
- The opposite side is the vertical distance between point A and the x-axis.
Let's calculate these lengths.
The horizontal side of the triangle is the x-coordinate of point A, given as -3.
The vertical side of the triangle is the y-coordinate of point A, given as 2.
Applying the Pythagorean theorem, we can find the length of the hypotenuse (h):
h^2 = (-3)^2 + 2^2
h^2 = 9 + 4
h^2 = 13
Therefore, the length of the hypotenuse (h) is the square root of 13.
Now, to find the exact value of cscθ, we need to determine the ratio of the hypotenuse to the opposite side, which can be expressed as:
cscθ = h / opposite side
Using the values we found, the exact value of cscθ is:
cscθ = √13 / 2
Thus, the exact value of cscθ is (√13)/2 or radical 13/2.
construct a triangle by drawing a perpendicular from the point (-3,2) to the x-axis.
by Pythagoras the hypotenuse is √(9+4) = √13
by definition cscθ = r/y = √13/2