Find the exact value of sec 3pi/2 If not defined, write undefined.
The Correct Answer and Explanation is:
To find the exact value of sec(3π2)\sec \left( \frac{3\pi}{2} \right)sec(23π), let’s break it down step by step.
Step 1: Understand the relationship between secant and cosine
The secant function is the reciprocal of the cosine function. In other words:sec(θ)=1cos(θ)\sec(\theta) = \frac{1}{\cos(\theta)}sec(θ)=cos(θ)1
So, to find sec(3π2)\sec \left( \frac{3\pi}{2} \right)sec(23π), we need to first find cos(3π2)\cos \left( \frac{3\pi}{2} \right)cos(23π).
Step 2: Find cos(3π2)\cos \left( \frac{3\pi}{2} \right)cos(23π)
The angle 3π2\frac{3\pi}{2}23π corresponds to 270° on the unit circle. At this angle, the point on the unit circle is at (0,−1)(0, -1)(0,−1). The cosine of an angle is the x-coordinate of the point on the unit circle, so:cos(3π2)=0\cos \left( \frac{3\pi}{2} \right) = 0cos(23π)=0
Step 3: Find sec(3π2)\sec \left( \frac{3\pi}{2} \right)sec(23π)
Now that we know cos(3π2)=0\cos \left( \frac{3\pi}{2} \right) = 0cos(23π)=0, we can substitute this into the equation for secant:sec(3π2)=1cos(3π2)=10\sec \left( \frac{3\pi}{2} \right) = \frac{1}{\cos \left( \frac{3\pi}{2} \right)} = \frac{1}{0}sec(23π)=cos(23π)1=01
Dividing by zero is undefined in mathematics. Therefore, we conclude:sec(3π2)=undefined\sec \left( \frac{3\pi}{2} \right) = \text{undefined}sec(23π)=undefined
Final Answer:
The exact value of sec(3π2)\sec \left( \frac{3\pi}{2} \right)sec(23π) is undefined.
Explanation:
The secant function relies on the cosine function, and when the cosine value is zero, the secant function becomes undefined. This is because division by zero is not a valid mathematical operation. In the case of 3π2\frac{3\pi}{2}23π, the cosine function equals zero, so the secant function is also undefined.
