Problem 66 - Contemporary Precalculus

Question

In Exercises \(65-70,\) draw a rough sketch to determine if the given number is positive. $$\cos 2$$

Step-by-Step Solution

Answer

Answer: No, the value of cos 2 is negative.

Step 1: Recall Cosine Function Properties

A cosine function is periodic with a period of \(2\pi\) and is symmetric about the y-axis. The cosine function is positive in the first and fourth quadrants and negative in the second and third quadrants.

Step 2: Determine the Quadrant of the 2 Radian Angle

First, we need to find out in which quadrant the angle of 2 radians lies. 2 radians is between \(\pi\) (3.14) and \(\frac{\pi}{2}\) (1.57). Therefore, the 2 radian angle lies in the second quadrant.

Step 3: Determine the Sign of Cosine Function

We know that the cosine function is negative in the second quadrant. Therefore, the value of \(\cos 2\) will be negative.

Step 4: Conclusion

We have determined that the given number \(\cos 2\) is negative, as it lies in the second quadrant where the cosine function is negative.

Key Concepts

Periodicity of Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent have a repeating pattern over a defined interval, termed as their 'periodicity.' For the cosine function, which is the focus of our exercise, the period is \(2\pi\). This implies that the cosine function repeats its values every \(2\pi\) radians.

Mathematically speaking, for any angle \(\theta\), the cosine function satisfies the property \(\cos(\theta+2\pi n) = \cos(\theta)\) for every integer \(n\). So, if you move around the unit circle by an angle of \(2\pi\), you would have made a full rotation back to your starting point, resulting in the same cosine value.

In practice, understanding the periodic nature of the cosine function can help students to solve problems that involve angles greater than \(2\pi\) or less than \(0\) by simply finding an equivalent angle within the first period \(0 \leq \theta < 2\pi\). This can make complex trigonometric problems more manageable and is precisely why recognizing the periodicity of trigonometric functions is crucial in exercises like determining the sign of \(\cos 2\).

Quadrants in Trigonometry

The coordinate plane is divided into four sections called quadrants, which are counterclockwise numbered I (first quadrant) to IV (fourth quadrant). Each quadrant contains angles with specific sign conventions for trigonometric functions.

In the first quadrant, all trigonometric functions are positive. Moving to the second quadrant, only sine and its reciprocal function, cosecant, remain positive. When we reach the third quadrant, tangent and its reciprocal function, cotangent, are the only positive ones. Finally, in the fourth quadrant, cosine and its reciprocal function, secant, are positive.

Determining the Quadrant of an Angle

When given an angle, like the 2 radian angle in our exercise, students must identify which quadrant it lies in to infer the sign of the trigonometric function. The 2 radian angle is greater than \(\frac{\pi}{2}\) but less than \(\pi\), placing it in the second quadrant. Knowing which trigonometric functions are positive or negative in each quadrant is essential for correctly solving trigonometry problems.

Properties of Cosine

The cosine function, one of the primary trigonometric functions, has several key properties that are relevant to our exercise. It is an even function, meaning that its graph is symmetric about the y-axis, which results in \(\cos(\theta) = \cos(-\theta)\). This property is especially useful for recognizing patterns in the function's behavior over its domain.

Another significant property is that the cosine function takes on values between -1 and 1, inclusive. This is because the range of cosine represents the x-coordinates of points on the unit circle, which are confined within this interval.

Cosine in Different Quadrants

The sign of the cosine function is determined by the quadrant in which the angle terminates. As mentioned previously, cosine is positive in the first and fourth quadrants and negative in the second and third quadrants. This guides us to conclude that \(\cos 2\), where the angle of 2 radians falls in the second quadrant, will be negative. Knowledge of these properties allows students to solve trigonometric equations and evaluate trigonometric expressions effectively.