Problem 20 - Algebra and Trigonometry

Question

let \(\theta\) be an angle in standard position. Name the quadrant in which \(\theta\) lies. $$ \tan \theta<0, \quad \sin \theta<0 $$

Step-by-Step Solution

Answer

Theta lies in Quadrant III.

Step 1: Understanding signs of the trigonometric functions

The sign of the functions \(\sin\) and \(\tan\) depend on the quadrant of the coordinates system. Recall that in Quadrant I both \(\sin\) and \(\tan\) are positive, in Quadrant II \(\sin\) is positive and \(\tan\) is negative, in Quadrant III both \(\sin\) and \(\tan\) are negative, and in Quadrant IV \(\sin\) is negative and \(\tan\) is positive.

Step 2: Identifying the quadrant where \(\sin\theta < 0\) and \(\tan\theta < 0\)

By analyzing the signs of the functions \(\sin\) and \(\tan\) in each quadrant, we can see that \(\sin\theta < 0\) and \(\tan\theta < 0\) are both valid in Quadrant III.

Key Concepts

Angle in Standard Position

When we talk about an angle in standard position, we are referring to a specific way of positioning an angle on a coordinate plane. This concept helps us easily identify the characteristics of the angle. Here's how it works:

• The vertex of the angle is at the origin of the coordinate plane.
• The initial side of the angle lies along the positive x-axis.
• The terminal side is determined by how far the angle rotates from the initial position.

Angles can be measured in degrees or radians, and they open counterclockwise for positive angles and clockwise for negative angles. This positioning is crucial for determining how different trigonometric functions behave in various quadrants of the plane.

Quadrants

The coordinate plane is divided into four regions, known as quadrants. These quadrants help us determine the sign of trigonometric functions, like sine and tangent, based on the angle's position:

• Quadrant I: Both sine and tangent functions are positive.
• Quadrant II: Sine is positive, while tangent is negative.
• Quadrant III: Both sine and tangent are negative.
• Quadrant IV: Sine is negative, and tangent is positive.

Understanding which quadrant an angle lies in allows us to predict the signs of these trigonometric functions. In our problem, \(\sin \theta < 0\) and \(\tan \theta < 0\) tell us that the angle \(\theta\) is in Quadrant III.

Sine Function

The sine function, denoted as \(\sin \theta\), is one of the primary trigonometric functions used to describe the properties of angles in the coordinate plane. It is defined as the ratio of the opposite side to the hypotenuse in a right triangle. In terms of the unit circle, it gives the y-coordinate of a point associated with an angle on the circle:

• The value of \(\sin \theta\) tells us how high the point is above or below the x-axis.
• In Quadrants I and II, \(\sin \theta\) is positive. In Quadrants III and IV, it is negative.
• A negative \(\sin \theta\) indicates that the terminal side of the angle is below the x-axis.

This function is vital for understanding wave patterns and oscillations, and it relates directly to our problem by indicating which quadrants have negative values for \(\sin \theta\).

Tangent Function

The tangent function, represented as \(\tan \theta\), is another foundational trigonometric function. It is defined as the ratio of the sine function to the cosine function (\[\tan \theta = \frac{\sin \theta}{\cos \theta}\]). In the context of a right triangle, it is the ratio of the opposite side to the adjacent side:

• In Quadrant I, \(\tan \theta\) is positive because both \(\sin \) and \(\cos \) are positive.
• In Quadrant II, \(\tan \theta\) is negative due to positive \(\sin \) and negative \(\cos \).
• In Quadrant III, \(\tan \theta\) becomes positive again as both \(\sin \) and \(\cos \) are negative.
• In Quadrant IV, it returns to negative with negative \(\sin \) and positive \(\cos \).

In our specific problem, a negative tangent function paired with a negative sine function confirms our angle \(\theta\) exists in Quadrant III.