Given tan theta=-8/5 and sin theta < 0, find cos theta.
Nahi samjha
To find cos theta, we can use the given information and the trigonometric identity relating tan theta to sin theta and cos theta.
1. Recall that tan theta is defined as the ratio of sin theta to cos theta.
tan theta = sin theta / cos theta
2. Substitute the given value of tan theta and sin theta in the equation.
-8/5 = sin theta / cos theta
3. Since sin theta is negative, we know that theta lies in either the third or fourth quadrant. In both of these quadrants, sin theta is negative.
4. We can assign a negative value to sin theta in the equation.
-8/5 = -sin theta / cos theta
5. Cross multiply the equation to eliminate the fractions.
(-8/5) * cos theta = -sin theta
6. Divide both sides of the equation by (-8/5) to solve for cos theta.
cos theta = (-sin theta) / (-8/5)
7. The negative signs cancel out, leaving us with:
cos theta = sin theta / (8/5)
So, cos theta is equal to sin theta divided by 8/5.
To find the value of cos theta, we can use the trigonometric identity:
cos² theta + sin² theta = 1
Given that sin theta < 0, we know that the value of sin theta is negative.
We are also given that tan theta equals -8/5, which is the ratio of the opposite side to the adjacent side in a right triangle.
From the given information, we can determine that the value of sin theta is -5 and the value of cos theta is 8.
Now, let's calculate the value of cos theta.
Using the Pythagorean Identity, we can rearrange the equation:
cos² theta = 1 - sin² theta
Plugging in the value of sin theta as -5:
cos² theta = 1 - (-5)²
cos² theta = 1 - 25
cos² theta = -24
Given that cos theta is a real number, the value of cos theta cannot be negative. Hence, there is no real solution for cos theta in this case.
if tanØ < 0 and sinØ < 0 , then Ø must be in quad IV
from tanØ = -8/5 in IV, we have x = 5, y = -8
r^2 = 25+64 = 89
r = √89
then cosØ = 5/√89