find the value of cotè given secè=5/4 and cscè<0
To find the value of cotθ, we need to first find the value of tanθ. We can start by using the given information about secθ and cscθ.
We are given secθ = 5/4 and cscθ < 0.
Recall that secθ is the reciprocal of cosθ and cscθ is the reciprocal of sinθ. Since secθ is positive (5/4 > 0), we know that cosθ is also positive.
Now, since cscθ is negative (cscθ < 0), we know that sinθ must also be negative.
We can use these two pieces of information to determine the quadrant in which θ lies.
In the unit circle, sinθ is negative in the third and fourth quadrants, and cosθ is positive in the first and fourth quadrants.
Since both sinθ and cosθ are negative in the fourth quadrant, θ must lie in the fourth quadrant.
In the fourth quadrant, tanθ is negative, and the reciprocal of tanθ is cotθ.
So, if we can find the value of tanθ, we can find the value of cotθ by taking its reciprocal.
Using the trigonometric identity tanθ = sinθ / cosθ, we can find the value of tanθ.
Since sinθ is negative, we need to find the negative value of sinθ in the fourth quadrant. We can use the Pythagorean identity sin^2θ + cos^2θ = 1, and since cosθ is positive (5/4 > 0), we can determine the value of sinθ.
Let's solve for sinθ using the Pythagorean identity:
sin^2θ + cos^2θ = 1
sin^2θ + (5/4)^2 = 1
sin^2θ + 25/16 = 1
sin^2θ = 1 - 25/16
sin^2θ = 16/16 - 25/16
sin^2θ = -9/16
Since sinθ is negative, we can take the negative square root:
sinθ = -√(-9/16) = -3/4 (since √(9) = 3)
Now that we know sinθ, we can use the equation tanθ = sinθ / cosθ:
tanθ = (-3/4) / (5/4) = -3/5
Finally, we can find cotθ by taking the reciprocal of tanθ:
cotθ = 1 / (tanθ) = 1 / (-3/5) = -5/3
Therefore, the value of cotθ is -5/3.