Problem 3 - Calculus

Question

Draw the angle using a ray through the origin, and determine whether the sine, cosine, and tangent of that angle are positive, negative, zero, or undefined. $$\frac{\pi}{4}$$

Step-by-Step Solution

Answer

All sine, cosine, and tangent values for \( \frac{\pi}{4} \) are positive.

Step 1: Understand the Angle

The angle given is \( \frac{\pi}{4} \), which is equivalent to 45 degrees. This angle is located in the first quadrant of the unit circle.

Step 2: Draw the Angle

To draw the angle \( \frac{\pi}{4} \), start from the positive x-axis (right horizontal axis) and measure 45 degrees counterclockwise. Draw a ray starting from the origin that makes a 45-degree angle with the positive x-axis.

Step 3: Determine Sine of the Angle

In the first quadrant, sine values are positive. For the angle \( \frac{\pi}{4} \), \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \), which is a positive number.

Step 4: Determine Cosine of the Angle

In the first quadrant, cosine values are also positive. For the angle \( \frac{\pi}{4} \), \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \), which is a positive number.

Step 5: Determine Tangent of the Angle

In the first quadrant, tangent values are positive. The tangent of \( \frac{\pi}{4} \) is calculated as the ratio of sine to cosine: \( \tan(\frac{\pi}{4}) = \frac{\sin(\frac{\pi}{4})}{\cos(\frac{\pi}{4})} = 1 \), which is a positive number.

Key Concepts

Unit Circle

The unit circle is a powerful tool in trigonometry, representing a circle with a radius of 1 centered at the origin of a coordinate plane. This circle allows us to visually interpret and calculate trigonometric functions for various angles. Trigonometric angles are measured starting from the positive x-axis and stretching counterclockwise around the circle. Since the radius is 1, any point on the circumference can serve as a terminal side of an angle, giving it a natural and easy way to explore the sine and cosine values for that angle.
The unit circle is divided into four quadrants, each of which corresponds to specific positive and negative signs for sine, cosine, and tangent functions. This division aids in understanding how trigonometric ratios change based on the angle's position.
Here’s what to remember about the unit circle:

• Sine values are represented by the y-coordinate of a point on the unit circle.
• Cosine values are represented by the x-coordinate of the same point.
• The tangent function, which is the ratio of sine to cosine, brings additional insight for each angle.

By studying the unit circle, we gain a comprehensive overview of how angles relate to trigonometric functions.

First Quadrant

The first quadrant of the unit circle is a particularly straightforward section to understand. It is the section where the angles range from 0 to 90 degrees, or from 0 to \( \frac{\pi}{2} \) in radians. This is where both the x (cosine) and y (sine) coordinates are positive.
When dealing with angles in the first quadrant, both the sine and cosine functions yield positive results. This is because any ray in this section comes from the origin, pointing to a point on the unit circle where both the sine and cosine values are positive. Consequently, the tangent function, which is the division of sine by cosine, is also positive.
Understanding the first quadrant helps ease the exploration of trigonometric functions:

• From 0 to \( \frac{\pi}{4} \) radians, both sine and cosine gradually increase.
• At \( \frac{\pi}{4} \), both sine and cosine are equal, which is why \( \tan(\frac{\pi}{4}) = 1 \).
• Sine and cosine values remain positive until you pass \( \frac{\pi}{2} \), at which point you enter the second quadrant.

This quadrant exemplifies the simplest scenario in understanding trigonometric properties due to its positive coordinate signs.

Positive and Negative Values of Trigonometric Functions

Understanding the positive and negative values of trigonometric functions is crucial for interpreting the unit circle. Each quadrant of the unit circle has characteristic signs for sine, cosine, and tangent functions, dictated by the x and y positional values.
In the first quadrant:

• Sine, cosine, and tangent functions are all positive as both coordinates are positive.
• This means for angles like \( \frac{\pi}{4} \), you're working with positive outcomes in trigonometric calculations.

As you move to different quadrants, these signs change:

• In the second quadrant, sine remains positive, cosine becomes negative, and tangent is negative.
• In the third quadrant, both sine and cosine are negative, making tangent positive again since it's the ratio of two negatives.
• In the fourth quadrant, sine becomes negative, while cosine remains positive, leading to a negative tangent.

These changes across the quadrants reflect how the unit circle naturally encodes trigonometric alternations of positivity and negativity as angles grow beyond the first quadrant. Recognizing these patterns allows us to accurately determine the values no matter the angle.