Problem 18 - Technical Mathematics with Calculus

Question

State whether the following expressions are positive or negative. Do not use your calculator, and try not to refer to your book. $$\tan 315^{\circ}$$

Step-by-Step Solution

Answer

\(\tan 315^{\circ}\) is negative.

Step 1: Understand Tangent's Sign in Different Quadrants

The sign of the tangent function depends on the quadrant in which the angle is located. Recall that tangent is positive in the first and third quadrants, and negative in the second and fourth quadrants.

Step 2: Determine the Quadrant for the Given Angle

Angle measurement of 315 degrees corresponds to the fourth quadrant since it is more than 270 degrees and less than 360 degrees.

Step 3: Determine the Sign of the Tangent

Since 315 degrees is in the fourth quadrant where tangent is negative, the expression \(\tan 315^{\circ}\) is negative.

Key Concepts

Understanding Tangent's Sign

The tangent function, one of the fundamental trigonometric functions, can sometimes pose a challenge when you're learning trigonometry, particularly when you need to determine the sign of its value without using a calculator. To grasp the concept of the tangent's sign, it's crucial to first understand that it is a ratio of the sine to the cosine of an angle (i.e., \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)).

The sign of the tangent function is directly influenced by the signs of both sine and cosine. Given that sine represents the y-coordinate, and cosine represents the x-coordinate on the unit circle, their signs change depending on the quadrant the terminal side of the angle lies in. The unit circle is divided into four quadrants:

• In the first quadrant, both sine and cosine are positive, making the tangent positive.
• In the second quadrant, sine is positive and cosine is negative, resulting in a negative tangent.
• In the third quadrant, both sine and cosine are negative, which makes the tangent positive once more.
• In the fourth quadrant, sine is negative and cosine is positive, leading to a negative tangent.

By simply knowing the quadrant in which the angle lies, you can determine the sign of the tangent without having to rely on a calculator.

Navigating the Angle Quadrants

An angle's measurement can tell you a great deal about its position relative to the coordinate axes, and specifically, in which quadrant it lies. Here's a quick guide:

• Angles measuring between 0 and 90 degrees are in the first quadrant.
• Angles between 90 and 180 degrees fall in the second quadrant.
• Third quadrant angles range from 180 to 270 degrees.
• Finally, angles from 270 to 360 degrees are situated in the fourth quadrant.

An understanding of angle quadrants is more than just an academic exercise; it's essential for interpreting the behavior of trigonometric functions. Memorizing the range of angles for each quadrant can be useful when you're trying to quickly determine the nature of a trigonometric function's value without a calculator. For example, recognizing that a 315-degree angle falls in the fourth quadrant immediately informs us about the signs of the sine, cosine, and tangent functions at that angle.

Mastering Trigonometry Without a Calculator

Being able to work with trigonometric functions without a calculator is a valuable skill, strengthening your understanding of the concepts and improving your ability to solve problems mentally. Here are a few tips:

• Memorize the unit circle, including the coordinates for commonly used angles such as 30, 45, 60, 90, etc.
• Understand the relationship between the trigonometric functions and the coordinates of points on the unit circle.
• Get comfortable with trigonometric identities that can simplify complex expressions without numerical computation.
• Practice visualizing the graphs of sine, cosine, and tangent to predict the function's behavior as the angle changes.

When faced with an exercise like determining whether \(\tan 315^{\regular{circ}}\) is positive or negative, these mental tools come into play. By knowing the quadrant and having a mental image of the unit circle, you can quickly arrive at the correct sign without the need for electronic aids. This method not only saves time during exams but also gives a deeper insight into the workings of trigonometric functions.