Suppose that $\theta$ is an angle in standard position whose terminal side intersects the unit circle at $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$. Find the exact values of $\cot \theta, \cos \theta$, and $\csc \theta$.
$\cot \theta=$
$\cos \theta=$
$\csc \theta=$
Answer (1)
The problem provides a point on the unit circle ( 2 2 , 2 2 ) corresponding to an angle θ .
We identify cos θ = 2 2 and sin θ = 2 2 directly from the coordinates.
We calculate cot θ as s i n θ c o s θ = 1 .
We calculate csc θ as s i n θ 1 = 2 .
cot θ = 1 , cos θ = 2 2 , csc θ = 2
Explanation
Problem Analysis We are given that the terminal side of angle θ intersects the unit circle at the point ( 2 2 , 2 2 ) . Our goal is to find the exact values of cot θ , cos θ , and csc θ .
Finding cos θ and sin θ Recall that for a point ( x , y ) on the unit circle corresponding to an angle θ , we have x = cos θ and y = sin θ . Therefore, we can directly identify the values of cos θ and sin θ from the given coordinates.
Values of cos θ and sin θ From the given point ( 2 2 , 2 2 ) , we have: cos θ = 2 2 sin θ = 2 2
Finding cot θ Now, we can find cot θ using the identity cot θ = s i n θ c o s θ .
cot θ = 2 2 2 2 = 1
Finding csc θ Next, we find csc θ using the identity csc θ = s i n θ 1 .
csc θ = 2 2 1 = 2 2 = 2 2 2 = 2
Final Answer Therefore, the exact values are: cot θ = 1 cos θ = 2 2 csc θ = 2
Examples
Understanding trigonometric functions and their values at specific points on the unit circle is crucial in many fields. For example, in physics, when analyzing simple harmonic motion, the position of an object can be described using sine and cosine functions. If you know the object's position at a certain time corresponds to a point on the unit circle, you can determine the angle and thus predict its future position. Similarly, in engineering, these concepts are used in signal processing and wave analysis to decompose complex signals into simpler sinusoidal components.
