Problem 27 - Algebra 2 and Trigonometry

Question

When \(\sin \theta=-1,\) find a value of: \(\quad\) a. \(\cos \theta \quad\) b. \(\tan \theta \quad\) c. \(\theta\)

Step-by-Step Solution

Answer

\( \cos \theta = 0 \); \( \tan \theta \) is undefined; \( \theta = \frac{3\pi}{2} + 2k\pi \).

Step 1: Understanding the Problem

We need to determine the values of \( \cos \theta \), \( \tan \theta \), and the angle \( \theta \) given that \( \sin \theta = -1 \). Recall that \( \sin \theta \) ranges from -1 to 1 and can be -1 when \( \theta \) is at specific angles on the unit circle.

Step 2: Identifying the Angle for \( \sin \theta = -1 \)

The sine function reaches -1 at \( \theta = \frac{3\pi}{2} \) radians, which corresponds to 270 degrees. At this angle, the point on the unit circle is \((0, -1)\).

Step 3: Finding \( \cos \theta \) at \( \frac{3\pi}{2} \)

When \( \theta = \frac{3\pi}{2} \), the point on the unit circle is \((0, -1)\). Therefore, \( \cos \theta = 0 \).

Step 4: Calculating \( \tan \theta \)

The tangent of an angle is the ratio of the sine to the cosine. At \( \theta = \frac{3\pi}{2} \), \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-1}{0} \). Since division by zero is undefined, \( \tan \theta \) is undefined at \( \theta = \frac{3\pi}{2} \).

Step 5: Conclusion

Therefore, when \( \sin \theta = -1 \), \( \cos \theta = 0 \), \( \tan \theta \) is undefined, and \( \theta = \frac{3\pi}{2} + 2k\pi \) where \( k \) is any integer.

Key Concepts

Understanding the Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin

• The coordinates of any point on the unit circle are representative of the cosine and sine of the angle formed by the line joining that point to the origin, with the positive x-axis.


• This means for a point (\(x, y\)), cosine is the x-coordinate and sine is the y-coordinate.


• For example, at \( \theta = \frac{3\pi}{2} \), the corresponding point is \( (0, -1) \), giving us \( \cos \theta = 0 \)and \( \sin \theta = -1 \).

The unit circle is immensely helpful because it allows us to understand how the sine and cosine functions behave not just within the standard 0 to \(2\pi \) radian interval, but by repeating these values every full circle, which is every \(2\pi \) radians.

Diving into the Sine Function

The sine function, denoted as \( \sin \theta \), is essential in trigonometry

• It measures the vertical coordinate of a point on the unit circle as \( \theta \) travels around it.


• Its function is periodic, meaning it repeats its values in regular intervals, every \(2\pi \) radians.

• The sine value ranges from -1 to 1, achieving -1 at \( \theta = \frac{3\pi}{2} \).


• This is critical as it can help define specific anglesbased on their sine values.

In real life, understanding this function supports multiple applications, such as modeling wave behaviors in physics. The rule that defines \( \sin \theta = -1 \) at \( \frac{3\pi}{2} \) is repeated every full rotation, since it is periodic.

Exploring the Tangent Function

The tangent function, represented as \(\tan \theta \), combines both the sine and cosine

• It is calculated using the formula \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).


• This tells us that tangent relates to the slope of the lineformed between a point \((x, y)\) and the origin on the unit circle.


• Tangent values can reach negative or positive infinity.


• \(\tan \theta\) becomes undefined whenever \(\cos \theta = 0 \), resulting in division by zero.This occurs at angles like \(\frac{3\pi}{2}\).

The function is highly useful in defining the angles of slopes and in calculating angles in various practical problems.

Understanding Radian Measure

Radian measure is an alternative to degree measure for angles

• One full circle is \(2\pi \) radians, which equals 360 degrees.


• Radian measure provides a direct way to relate the angle to the arc length it subtends.

• For example, \(\theta = \frac{3\pi}{2} \) is equivalent to 270 degrees, representing three-quarters of a full rotation.


• It is the preferred unit in mathematics because its properties simplify many formulas, making it possible to see trigonometric functions in terms of the unit circle.

Grasping the concept of radians is essential in transitioning smoothly between different angle representations,thus enhancing your problem-solving skills in trigonometry.