Problem 2 - Algebra and Trigonometry

Question

The sign of a trigonometric function of \(\theta\) depends on the _____ in which the terminal side of the angle \(\theta\) lies. In Quadrant II, \(\sin \theta\) is _____ (positive/negative). In Quadrant III, \(\cos \theta\) is _____ (positive/negative). In Quadrant IV, \(\sin \theta\) is _____ (positive/negative).

Step-by-Step Solution

Answer

The sign of a trigonometric function of \(\theta\) depends on the quadrant in which \(\theta\) lies. In Quadrant II, \(\sin \theta\) is positive. In Quadrant III, \(\cos \theta\) is negative. In Quadrant IV, \(\sin \theta\) is negative.

Step 1: Identify the Quadrants and their Characteristics

The coordinate plane is divided into four quadrants. Let's recall the trigonometric function signs in each quadrant: In Quadrant I, all trigonometric functions are positive. In Quadrant II, sine is positive while cosine and tangent are negative. In Quadrant III, tangent is positive while sine and cosine are negative. In Quadrant IV, cosine is positive while sine and tangent are negative.

Step 2: Determine the Dependence of the Function Signs

The sign of a trigonometric function of \(\theta\) depends on the quadrant in which the terminal side of the angle \(\theta\) lies.

Step 3: Evaluate the Sine Function in Quadrant II

In Quadrant II, the sine function \(\sin \theta\) is positive, because in this quadrant, the y-values (opposite side for sine) are positive.

Step 4: Evaluate the Cosine Function in Quadrant III

In Quadrant III, the cosine function \(\cos \theta\) is negative, as the x-values (adjacent side for cosine) are negative in this quadrant.

Step 5: Evaluate the Sine Function in Quadrant IV

In Quadrant IV, the sine function \(\sin \theta\) is negative, because in this quadrant, the y-values (opposite side for sine) are negative.

Key Concepts

Quadrants

The concept of quadrants is essential in trigonometry and geometry. When working with angles, the coordinate plane is divided into four unique sections. Each of these sections, known as quadrants, determines certain properties of trigonometric functions.

• **Quadrant I**: Here, both x and y coordinates are positive. Thus, all trigonometric functions (sine, cosine, and tangent) are positive.
• **Quadrant II**: In this quadrant, x is negative and y is positive. Only the sine function remains positive.
• **Quadrant III**: Both coordinates, x and y, become negative. Here, the tangent function is positive.
• **Quadrant IV**: The x coordinate returns to being positive while y is negative. Cosine is the only function that's positive here.

Understanding the coordinate signs in each quadrant helps in deciding the signs of trigonometric functions related to angles falling within those quadrants.

Sine Function

The sine function is a fundamental trigonometric function represented by \(\sin \theta\). It is based on the y-coordinate, which is equivalent to the vertical side of a right triangle in a circle with radius 1 (unit circle).

When dealing with quadrants:

• In **Quadrant I**, \(\sin \theta\) is positive because both x and y coordinates are positive. The angle lies in the first 90 degrees of the circle.
• In **Quadrant II**, \(\sin \theta\) remains positive. This is because the y-coordinate remains positive even though x becomes negative.
• In **Quadrant III**, \(\sin \theta\) turns negative as both x and y coordinates become negative.
• In **Quadrant IV**, the function \(\sin \theta\) remains negative because the y-coordinate is negative.

These characteristics of the sine function are crucial for solving various trigonometric problems involving angles and triangles.

Cosine Function

Cosine, represented as \(\cos \theta\), is another fundamental trigonometric function which is tied to the x-coordinate on a unit circle.

Its behavior in the quadrants is as follows:

• In **Quadrant I**, \(\cos \theta\) is positive as both x and y are positive.
• Moving to **Quadrant II**, \(\cos \theta\) becomes negative since x is negative, though y is still positive.
• In **Quadrant III**, \(\cos \theta\) remains negative because both x and y turn negative.
• Lastly, in **Quadrant IV**, \(\cos \theta\) is positive again since x reverts to positive values.

The understanding of the cosine function within each quadrant is helpful in graphing functions and understanding the transformation of angles across the coordinate plane.

Sign of Angles

The sign of an angle's trigonometric function is determined fundamentally by the quadrant in which its terminal side resides. This is because each quadrant imparts a different sign to the x and y coordinates.

To remember the signs for sine and cosine across the four quadrants:

• **Quadrant I**: All functions \(\sin \theta\), \(\cos \theta\) are positive.
• **Quadrant II**: Here, \(\sin \theta\) is positive, while \(\cos \theta\) is negative.
• **Quadrant III**: Both \(\sin \theta\) and \(\cos \theta\) are negative.
• **Quadrant IV**: \(\cos \theta\) is positive, and \(\sin \theta\) is negative.

Remembering these consistent sign patterns for trigonometric functions in each quadrant can simplify complex calculations and help in comprehending the geometric representation of angles.