Let the (-7,-4) be a point on the terminal side of theta find the exact values of cos theta,cosec theta and tan theta
To find the exact values of cosine (cos), cosecant (csc), and tangent (tan) for a given point on the terminal side of an angle theta, we can use the given coordinates of the point and apply the definitions of these trigonometric functions.
Let's start by labeling the coordinates of the point (-7, -4) as x and y:
x = -7
y = -4
We can find the radius (r) using the Pythagorean theorem: r^2 = x^2 + y^2
r^2 = (-7)^2 + (-4)^2
r^2 = 49 + 16
r^2 = 65
r ≈ √65
Now, we can find the cosine (cos(theta)), cosecant (csc(theta)), and tangent (tan(theta)) using the given coordinates:
cos(theta) = x/r = -7/√65
csc(theta) = r/y = √65/-4
tan(theta) = y/x = -4/-7
Therefore, the exact values of cos(theta), csc(theta), and tan(theta) are:
cos(theta) = -7/√65
csc(theta) = -√65/4 (or √65/-4, considering the reciprocal)
tan(theta) = 4/7
To find the exact values of cos theta, cosec theta, and tan theta given that (-7, -4) is a point on the terminal side of theta, we need to use the Pythagorean identity to determine the hypotenuse and the values of the adjacent and opposite sides.
The Pythagorean identity is as follows:
r^2 = x^2 + y^2
where r is the radius or hypotenuse, x is the adjacent side, and y is the opposite side.
Using the point (-7, -4) as x and y coordinates, we can find the radius (r):
r^2 = (-7)^2 + (-4)^2
r^2 = 49 + 16
r^2 = 65
r = √65
Now let's calculate the values of cos theta, cosec theta, and tan theta:
cos theta = x / r = -7 / √65
cosec theta = r / y = √65 / -4
tan theta = y / x = -4 / -7 = 4 / 7
Therefore, the exact values of cos theta, cosec theta, and tan theta are:
cos theta = -7 / √65
cosec theta = √65 / -4
tan theta = 4 / 7