Question
What is the largest value of cosine of θ
on the unit circle? What is the smallest value of cosine of θ
?
on the unit circle? What is the smallest value of cosine of θ
?
Answers
GPT-4o mini
On the unit circle, the cosine of an angle \( \theta \) corresponds to the x-coordinate of the point on the circle. The unit circle is defined by the equation \( x^2 + y^2 = 1 \).
The largest value of cosine \( \cos(\theta) \) occurs when the angle \( \theta \) is 0 radians (or any integral multiple of \( 2\pi \) radians), which gives:
\[
\cos(0) = 1
\]
The smallest value of cosine occurs when the angle \( \theta \) is \( \pi \) radians (or any odd integral multiple of \( \pi \) radians), which gives:
\[
\cos(\pi) = -1
\]
Thus:
- The largest value of \( \cos(\theta) \) is \( 1 \).
- The smallest value of \( \cos(\theta) \) is \( -1 \).
The largest value of cosine \( \cos(\theta) \) occurs when the angle \( \theta \) is 0 radians (or any integral multiple of \( 2\pi \) radians), which gives:
\[
\cos(0) = 1
\]
The smallest value of cosine occurs when the angle \( \theta \) is \( \pi \) radians (or any odd integral multiple of \( \pi \) radians), which gives:
\[
\cos(\pi) = -1
\]
Thus:
- The largest value of \( \cos(\theta) \) is \( 1 \).
- The smallest value of \( \cos(\theta) \) is \( -1 \).