Let (3,-2) be a point on the terminal side of theta. Find cos theta, csc theta, tan theta?
To find cos theta, csc theta, and tan theta, we need to determine the values of the trigonometric ratios based on the given point (3, -2).
First, let's find the hypotenuse of the right triangle formed by the given point (3, -2):
The hypotenuse can be found using the Pythagorean theorem:
hypotenuse^2 = x^2 + y^2
Substituting the given values, we have:
hypotenuse^2 = 3^2 + (-2)^2
hypotenuse^2 = 9 + 4
hypotenuse^2 = 13
Taking the square root of both sides gives us:
hypotenuse = √13
Now, let's calculate the trigonometric ratios:
cos theta = adjacent side / hypotenuse
cos theta = 3 / √13
To rationalize the denominator, we can multiply both the numerator and denominator by √13:
cos theta = 3√13 / 13
csc theta = hypotenuse / opposite side
csc theta = √13 / -2
To simplify the ratio, we can multiply both the numerator and denominator by -1:
csc theta = -√13 / 2
tan theta = opposite side / adjacent side
tan theta = -2 / 3
So, the values of the trigonometric ratios for theta, based on the given point (3, -2), are:
cos theta = 3√13 / 13
csc theta = -√13 / 2
tan theta = -2 / 3
To find the values of cosine (cos), cosecant (csc), and tangent (tan) of an angle theta, we need to use the coordinates of the point (3, -2) to determine the values of the trigonometric ratios.
First, let's visualize the point (3, -2) on the coordinate plane:
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(3,-2)
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From this point, we can determine the lengths of the two sides of the triangle formed by the point and the origin (0,0): the horizontal distance (adjacent side) is 3 units, and the vertical distance (opposite side) is 2 units.
To find the hypotenuse of the triangle, we can use the Pythagorean theorem:
hypotenuse^2 = adjacent^2 + opposite^2
Let's plug in the values:
hypotenuse^2 = 3^2 + (-2)^2 = 9 + 4 = 13
Taking the square root of both sides gives us the length of the hypotenuse:
hypotenuse = sqrt(13)
Now we can determine the values of the trigonometric ratios:
1. Cosine (cos):
cos(theta) = adjacent / hypotenuse = 3 / sqrt(13)
2. Cosecant (csc):
csc(theta) = hypotenuse / opposite = sqrt(13) / 2
3. Tangent (tan):
tan(theta) = opposite / adjacent = -2 / 3
So, the values of cos(theta), csc(theta), and tan(theta) for the given point (3, -2) are:
cos(theta) = 3 / sqrt(13)
csc(theta) = sqrt(13) / 2
tan(theta) = -2 / 3
is the origin at the vertex, and is the angle measured from the x axis?
then the distance from the origin to the point is sqrt(9+4)=sqrt13
sinTheta= -2/sqrt13 cosineTheta= 3/sqrt13
draw the diagrams.