(b) Find the following. Use exact values and not decimal approximations.
[tex]
\begin{array}{l}
\cos \left(\frac{\pi}{3}\right)=\square \\
\cos \left(-\frac{\pi}{3}\right)=
\end{array}
[/tex]
(c) Choose the correct statement.
* For any angle [tex]$\theta$[/tex], we have [tex]$\cos (-\theta)=-\cos \theta$[/tex]. (So cosine is an odd function.)
* For any angle [tex]$\theta$[/tex], we have [tex]$\cos (-\theta)=\cos \theta$[/tex]. (So cosine is an even function.)
* Neither of the above answers are true. (So cosine is neither an odd nor an even function.)
Answer (1)
cos ( 3 π ) = 2 1 .
cos ( − 3 π ) = 2 1 .
Cosine is an even function because cos ( − θ ) = cos ( θ ) .
The correct statement is: For any angle θ , we have cos ( − θ ) = cos θ . (So cosine is an even function.) 2 1
Explanation
Problem Analysis We are asked to find the exact values of cos ( 3 π ) and cos ( − 3 π ) , and then determine whether the cosine function is even, odd, or neither.
Calculate cos ( 3 π ) First, we need to find the exact value of cos ( 3 π ) . Recall that 3 π radians is equal to 6 0 ∘ . From the unit circle or special right triangles, we know that cos ( 6 0 ∘ ) = 2 1 . Therefore, cos ( 3 π ) = 2 1 .
Calculate cos ( − 3 π ) Next, we need to find the exact value of cos ( − 3 π ) . Since cosine is an even function, cos ( − θ ) = cos ( θ ) for any angle θ . Therefore, cos ( − 3 π ) = cos ( 3 π ) = 2 1 .
Determine if cosine is even, odd, or neither Finally, we need to determine whether cosine is an even function, an odd function, or neither. Since cos ( − θ ) = cos ( θ ) , cosine is an even function.
Conclusion Therefore, cos ( 3 π ) = 2 1 , cos ( − 3 π ) = 2 1 , and cosine is an even function.
Examples
Understanding the properties of trigonometric functions like cosine is crucial in many fields. For instance, in physics, when analyzing simple harmonic motion, the cosine function describes the position of an object oscillating over time. Knowing that cosine is an even function, i.e., cos ( − θ ) = cos ( θ ) , helps predict the object's position at symmetric points in time relative to the center of oscillation. This symmetry simplifies calculations and provides deeper insights into the system's behavior, making it easier to model and control oscillating systems in engineering and physics.
