Problem 17 - Precalculus

Question

Indicate the quadrant in which the terminal side of \(\theta\) must lie in order for the information to be true. \(\tan \theta\) is negative and \(\sin \theta\) is positive.

Step-by-Step Solution

Answer

The terminal side of \(\theta\) lies in the second quadrant.

Step 1: Understanding the Trigonometric Ratios

To determine which quadrant \(\theta\) lies in, we need to know where each trigonometric ratio (\(\tan \, \theta\) and \(\sin \, \theta\)) is positive or negative. The tangent function \(\tan \theta\) is negative in the second and fourth quadrants. The sine function \(\sin \theta\) is positive in the first and second quadrants.

Step 2: Identifying Overlapping Quadrants

Since \(\tan \theta\) is negative and \(\sin \theta\) is positive, \(\theta\) must lie in a quadrant where the tangent is negative and sine is positive. From the previous step, \(\tan \theta\) is negative in the second and fourth quadrants, while \(\sin \theta\) is positive in the first and second quadrants. The only overlapping quadrant is the second quadrant.

Step 3: Conclusion of the Correct Quadrant

Therefore, the terminal side of \(\theta\) lies in the second quadrant where \(\tan \theta\) is negative and \(\sin \theta\) is positive.

Key Concepts

Trigonometric Ratios

When studying trigonometry, one of the foundational concepts is understanding trigonometric ratios. These ratios involve relationships between the angles and sides of a right triangle.

• Sine (\(\sin \theta\)): Represents the ratio of the length of the opposite side to the hypotenuse.
• Cosine (\(\cos \theta\)): Represents the ratio of the length of the adjacent side to the hypotenuse.
• Tangent (\(\tan \theta\)): Represents the ratio of the length of the opposite side to the adjacent side.

Understanding these ratios helps in determining the angles and sides of triangles, as well as identifying the behavior of various trigonometric functions. Each of these ratios behaves differently in each of the four quadrants of the unit circle, which is important for solving trigonometric equations.

Tangent Function

The tangent function, denoted as \(\tan \theta\), plays a vital role in trigonometry. It is defined as the ratio of the sine of an angle to the cosine of an angle: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
The behavior of the tangent function varies depending on the quadrant:

• In the first quadrant, \(\tan \theta\) is positive because both \(\sin \theta\) and \(\cos \theta\) are positive.
• In the second quadrant, \(\tan \theta\) is negative because \(\sin \theta\) is positive and \(\cos \theta\) is negative.
• In the third quadrant, \(\tan \theta\) is positive since both \(\sin \theta\) and \(\cos \theta\) are negative.
• In the fourth quadrant, \(\tan \theta\) is negative because \(\sin \theta\) is negative and \(\cos \theta\) is positive.

Knowing where the tangent is positive or negative helps in identifying the angle's position within the coordinate plane.

Sine Function

The sine function, represented by \(\sin \theta\), is crucial in the study of trigonometry. It measures the vertical component of an angle, specifically the ratio of the opposite side to the hypotenuse in a right triangle.
The values of the sine function change as the angle \(\theta\) moves through the quadrants of the unit circle:

• In the first quadrant, \(\sin \theta\) is positive.
• In the second quadrant, \(\sin \theta\) is also positive, as it is the height of the unit circle above the x-axis.
• In the third quadrant, \(\sin \theta\) becomes negative because it falls below the x-axis.
• In the fourth quadrant, \(\sin \theta\) is also negative.

Understanding where the sine of an angle is positive or negative is key to solving problems related to trigonometric functions and locating the angle in the correct quadrant.