Problem 100 - Precalculus

Question

Determine whether each statement is true or false. The trigonometric function value for any angle with positive measure must be positive.

Step-by-Step Solution

Answer

The statement is false; trigonometric functions can be negative for positive angles.

Step 1: Understanding the Problem

We need to determine if it is true that the value of a trigonometric function for any angle with a positive measure is always positive. Trigonometric functions include sine, cosine, and tangent, among others.

Step 2: Consider the Unit Circle

On the unit circle, an angle is measured from the positive x-axis counterclockwise. The x-coordinate corresponds to the cosine of the angle, and the y-coordinate corresponds to the sine of the angle. Angles can be positive, and we will check the signs of their sine and cosine values.

Step 3: Analyze the Quadrants

Positive angles can fall into the first quadrant (0 to 90 degrees), second quadrant (90 to 180 degrees), third quadrant (180 to 270 degrees), or fourth quadrant (270 to 360 degrees). In the second quadrant, cosine is negative; in the third and fourth quadrants, both sine and cosine can be negative.

Step 4: Determine the Truth Value

Since in the second quadrant, cosine is negative, and in the third and fourth quadrants, both sine and cosine can be negative, the statement that the value of any trigonometric function with a positive angle must be positive is false.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. The primary reason it is used in trigonometry is that it helps to define the trigonometric functions for all angles. When an angle is drawn in the unit circle from the positive x-axis, the endpoint of the ray falls on the circle. The coordinates of this endpoint can be used to find the sine (\(\sin\theta\)) and cosine (\(\cos\theta\)) of the angle.

• The x-coordinate represents the cosine of the angle.
• The y-coordinate represents the sine of the angle.

By using the unit circle, we can easily understand how these trigonometric functions behave as the angle varies.

Sine and Cosine

The sine and cosine functions are two of the most important trigonometric functions. They derive their names from the unit circle usage as they are directly linked to the coordinates

• Sine corresponds to the y-coordinate of a point on the unit circle.
• Cosine corresponds to the x-coordinate of a point on the unit circle.

These functions allow us to determine properties of angles and the relationships in triangles, especially right-angled triangles. Importantly, sine and cosine functions are periodic with a period of 360 degrees or \(2\pi\) radians, meaning their values repeat every full circle around the unit circle.

Angle Measurement

In trigonometry, angles can be measured in degrees or radians. Understanding angle measurement is crucial for working with trigonometric functions and the unit circle. The conversion between these two units is essential:

• 1 full circle = 360 degrees = \(2\pi\) radians
• To convert degrees to radians, multiply by \(\frac{\pi}{180}\).
• To convert radians to degrees, multiply by \(\frac{180}{\pi}\).

Knowing how to measure angles correctly ensures you apply the trigonometric functions correctly, especially when dealing with different quadrants on the unit circle.

Quadrants

The coordinate plane is divided into four quadrants, each affecting the signs of trigonometric functions for given angles. Understanding the quadrants allows you to determine the signs of sine and cosine based on the given angle:

• First Quadrant (0 to 90 degrees): Both sine and cosine values are positive.
• Second Quadrant (90 to 180 degrees): Sine is positive, but cosine is negative.
• Third Quadrant (180 to 270 degrees): Both sine and cosine values are negative.
• Fourth Quadrant (270 to 360 degrees): Sine is negative, and cosine is positive.

These properties prove useful when resolving exercises in trigonometry, such as determining whether a trigonometric function's value is positive or negative based on the angle's position on the unit circle.