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

Answer 1

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