Problem 42 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Use the interactive unit circle tool at BigIdeasMath.com to describe all values of \(\theta\) for each situation. a. \(\sin \theta>0, \cos \theta<0\), and \(\tan \theta>0\) b. \(\sin \theta>0, \cos \theta<0\), and \(\tan \theta<0\)

Step-by-Step Solution

Answer

For case a) \(\theta\) lies in the second quadrant. For case b) there is no valid value for \(\theta\).

Step 1: Analyze case a)

In case a) \(\sin \theta>0, \cos \theta<0\), and \(\tan \theta>0\). From the rule 'All Students Take Calculus', positive sine values occur in 1st and 2nd quadrants, but cosine is negative in 2nd and 3rd quadrants. Since both sine and cosine values comply only in the 2nd quadrant, and tangent is a ratio of sine over cosine, then tangent will also be positive. Therefore, for a) \(\theta\) lies in the second quadrant.

Step 2: Analyze case b)

In case b) \(\sin \theta>0, \cos \theta<0\), and \(\tan \theta<0\). The situation for sine and cosine is the same as in a) which means \(\theta\) should be in the 2nd quadrant. But since tangent is the ratio of sine over cosine and the requirement now is that \(\tan \theta<0\), we have a conflict. It's not possible for \(\theta\) to be in the second quadrant because tangent will be positive there. This indicates that there is no valid value for \(\theta\) in this case.

Key Concepts

Unit Circle

The unit circle is a powerful tool for understanding trigonometric functions. It's a circle with a radius of 1 centered at the origin of a coordinate plane.
The angle \( \theta \) represents the rotation from the positive x-axis:

• Positive angles result from counterclockwise rotation.
• Negative angles result from clockwise rotation.

The unit circle helps visualize the relationships between angles and the sine, cosine, and tangent values.
Each point on the unit circle corresponds to \((\cos \theta, \sin \theta)\), helping you see which quadrants meet certain conditions.

Sine Function

The sine function assigns to each angle \( \theta \) the y-coordinate of the corresponding point on the unit circle.

• This means \( \sin \theta \) is positive in the first and second quadrants.
• Sine reaches its maximum value of 1 at \( 90^\circ \) or \( \frac{\pi}{2} \).
• It descends to 0 at \( 180^\circ \) or \( \pi \).

Understanding where sine is positive or negative helps in solving equations and understanding properties of waves or oscillations.

Cosine Function

The cosine function gives the x-coordinate on the unit circle for angle \( \theta \).
This means:

• \( \cos \theta \) is positive in the first and fourth quadrants.
• Cosine is zero at \( 90^\circ \) or \( \frac{\pi}{2} \).
• It becomes negative in the second and third quadrants.

Cosine’s relationship with sine through the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) links these functions strongly, which is quite useful in trigonometry.

Tangent Function

The tangent function is defined as the ratio of sine to cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
This means:

• Tangent is positive when both sine and cosine are positive (1st quadrant) or both are negative (3rd quadrant).
• Tangent is negative in the second quadrant, i.e., positive sine and negative cosine.

Tangent’s undefined values occur when cosine is zero (e.g., \( 90^\circ \) or \( 270^\circ \)).
This function is useful for understanding angles of inclination and slopes.