Problem 1 - Algebra 2

Question

Find the value of each expression. $$ \tan \theta, \text { if } \sin \theta=\frac{1}{2} ; 90^{\circ} \leq \theta<180^{\circ} $$

Step-by-Step Solution

Answer

\(\tan \theta = -\frac{\sqrt{3}}{3}\).

Step 1: Understand the Given Information

We are given that \(\sin \theta = \frac{1}{2}\) and that \( 90^{\circ} \leq \theta < 180^{\circ} \). This specifies that the angle \( \theta \) is in the second quadrant of the unit circle.

Step 2: Determine the Reference Angle

Since \( \sin \theta = \frac{1}{2} \), the reference angle \( \alpha = 30^{\circ} \) because \( \sin(30^{\circ}) = \frac{1}{2} \). In the second quadrant, the angle \( \theta \) is represented as \( 180^{\circ} - \alpha = 150^{\circ} \).

Step 3: Determine the Tangent

In the second quadrant, \( \tan \theta \) is negative. Using \( \tan(150^{\circ}) \), we have \( \tan 150^{\circ} = \tan(180^{\circ} - 30^{\circ}) = -\tan 30^{\circ} = -\frac{1}{\sqrt{3}} \).

Step 4: Simplify the Result

The value of \( \tan 30^{\circ} \) is \( \frac{1}{\sqrt{3}} \) which simplifies to \( \frac{\sqrt{3}}{3} \), hence \( \tan(150^{\circ}) = -\frac{\sqrt{3}}{3} \). Therefore, \( \tan \theta = -\frac{\sqrt{3}}{3} \).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that simplifies understanding angles and their trigonometric functions. Picture a circle with a radius of 1, centered at the origin of a coordinate plane. This is the unit circle, a handy tool because it lets us directly relate an angle's measure to points on the circle.

• When you mark an angle on the unit circle, its terminal side intersects the circle at a specific point.
• The x-coordinate of this point represents the cosine of the angle, while the y-coordinate represents the sine.
• These relationships are pivotal: for any angle \(\theta\), \(\cos \theta\) and \(\sin \theta\) can be visualized.

Using the unit circle becomes essential when solving problems like determining points and values such as sine or cosine for specific angles.

Quadrants in Trigonometry

Trigonometry divides the coordinate plane into four quadrants, helping to understand how trigonometric functions behave across different angle ranges. Each quadrant represents a 90-degree section.

• First Quadrant (0° to 90°): Here, both sine and cosine values are positive, so functions like tangent are positive, too.
• Second Quadrant (90° to 180°): Sine remains positive, but cosine and tangent values become negative. This is critical since the exercise has \(90^{\circ} \leq \theta < 180^{\circ}\).
• Third Quadrant (180° to 270°): Both sine and cosine are negative, making tangent positive again, as it is the ratio of sine over cosine.
• Fourth Quadrant (270° to 360°): Sine is negative, cosine is positive, and tangent is negative.

Understanding which quadrant an angle falls into is essential for determining the sign (positive or negative) of trigonometric functions. In our problem, since \(\theta\) is in the second quadrant, the tangent of the angle is negative.

Reference Angle

The concept of a reference angle helps determine trigonometric values in any quadrant based on an angle's distance from the nearest x-axis. A reference angle is always between 0° and 90°.
To find a reference angle:

• In the first quadrant, the angle itself is the reference angle.
• In the second quadrant, subtract the angle from 180°. For example, an angle of \(150^{\circ}\) has a reference angle of \(180^{\circ} - 150^{\circ} = 30^{\circ}\).
• In the third quadrant, subtract 180° from the angle. For instance, \(210^{\circ}\) gives us \(210^{\circ} - 180^{\circ} = 30^{\circ}\).
• In the fourth quadrant, subtract the angle from 360°. So, a \(330^{\circ}\) angle translates to \(360^{\circ} - 330^{\circ} = 30^{\circ}\).

Reference angles are identical for any standard angle that shares the same terminal side, allowing simplification of trigonometric calculations as seen with \(\theta = 150^{\circ}\).