Let's find the value of cosine of 15 pi. To solve this, we need to use the periodic property of the cosine function. The cosine function has a period of 2 pi, which means cosine of theta plus 2 n pi equals cosine of theta for any integer n. This graph shows how the cosine function repeats every 2 pi units.
Now let's simplify 15 pi. We can write 15 pi as 14 pi plus pi. Since 14 pi equals 7 times 2 pi, it's a multiple of the period 2 pi. This means cosine of 15 pi equals cosine of 14 pi plus pi, which simplifies to cosine of pi. The number line shows how 15 pi and pi are equivalent positions on the cosine function.
Now we need to evaluate cosine of pi. We know that cosine of 15 pi equals cosine of pi. On the unit circle, pi radians corresponds to 180 degrees. At this angle, we reach the point negative 1, 0 on the circle. Since cosine gives us the x-coordinate of the point on the unit circle, cosine of pi equals negative 1. Therefore, cosine of 15 pi equals negative 1.
Let's summarize our solution. We started with cosine of 15 pi. We rewrote 15 pi as 14 pi plus pi, where 14 pi equals 7 times 2 pi. Since the cosine function has a period of 2 pi, cosine of 15 pi equals cosine of pi. And cosine of pi equals negative 1. Therefore, our final answer is cosine of 15 pi equals negative 1.
Let's verify our answer and look at similar examples. We can confirm that cosine of 15 pi equals cosine of 7 times 2 pi plus pi, which equals cosine of pi, which is negative 1. This pattern extends to other odd multiples of pi. For instance, cosine of 13 pi, 17 pi, and 21 pi all equal negative 1. The general pattern is that cosine of any odd multiple of pi equals negative 1, as shown by the red points on the graph.