Problem 43 - Algebra 2 and Trigonometry

Question

In \(28-43,\) for each function value, if \(0^{\circ} \leq \theta <3 60^{\circ},\) find, to the nearest degree, two values of \(\theta\) \(\tan \theta=0\)

Step-by-Step Solution

Answer

\( \theta = 0^{\circ} \) and \( \theta = 180^{\circ} \).

Step 1: Understand the Problem

We need to find two angles, \( \theta \), where the tangent of the angle is zero and \( 0^{\circ} \leq \theta < 360^{\circ} \).

Step 2: Tangent Function Analysis

Recall that \( \tan \theta = 0 \) when \( \sin \theta = 0 \) and \( \cos \theta eq 0 \). The basic angles where tangent is zero are at the standard angles where sine is zero: \( 0^{\circ} \) and \( 180^{\circ} \).

Step 3: Determine Possible Angles

Given that \( 0^{\circ} \leq \theta < 360^{\circ} \), the standard position angles where \( \tan \theta = 0 \) are \( \theta = 0^{\circ} \) and \( \theta = 180^{\circ} \).

Step 4: Check Within Given Interval

Verify if the obtained angles \( 0^{\circ} \) and \( 180^{\circ} \) fall within the interval \( 0^{\circ} \leq \theta < 360^{\circ} \). Both angles are valid since they satisfy the condition.

Step 5: Conclusion

The two angles where \( \tan \theta = 0 \) within the interval \( 0^{\circ} \leq \theta < 360^{\circ} \) are \( \theta = 0^{\circ} \) and \( \theta = 180^{\circ} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. They are incredibly useful in simplifying trigonometric expressions and solving equations. These identities form the backbone of understanding how trigonometric functions like sine, cosine, and tangent relate to each other.
For the tangent function, which is the ratio of the sine and cosine:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
• This means that whenever the sine of an angle is zero, the tangent of that angle is also zero, as long as the cosine is not zero.

Several key identities are often used in conjunction with tangent, such as:

• \( \tan(\theta + 180^\circ) = \tan(\theta) \), indicating its periodicity.
• Fundamental identity: \( \sin^2 \theta + \cos^2 \theta = 1 \), which ties back into the behavior of sine and cosine.

Understanding these identities allows us to predict the behavior of trigonometric functions across different angles and solve more complex problems effectively.

Angle Measurement

Measuring angles is essential to understanding position and direction in trigonometry. Angles can be measured in degrees or radians, but here we'll focus on degrees, which is more intuitive for beginners.
Degrees can range from \(0^\circ\) to \(360^\circ\) for a full circle. This range is useful because it represents a full rotation around a circle. For this specific exercise, we are considering angles where \(0^\circ \leq \theta < 360^\circ \).
In this context, finding angles where certain trigonometric functions yield specific results focuses on standard angles. Standard angles are common benchmarks like \(0^\circ\), \(90^\circ\), \(180^\circ\), and \(270^\circ\). These angles have simple sine, cosine, and tangent values, making calculations easier.

• For instance, \( \tan 0^\circ = 0 \) because \( \sin 0^\circ = 0 \) and \( \cos 0^\circ = 1 \).
• Similarly, \( \tan 180^\circ = 0 \) because \( \sin 180^\circ = 0 \) and \( \cos 180^\circ = -1 \).

By understanding these standard angles and how they relate to the trigonometric functions, we can accurately predict and calculate the behavior of these functions within specified intervals.

Unit Circle

The unit circle is a fundamental concept in trigonometry that helps in understanding the behavior of trigonometric functions. It is a circle with a radius of one unit, centered at the origin of a coordinate plane.
The unit circle allows for a geometric representation of angles and trigonometric functions. When we say \( \tan \theta = 0 \), we can visualize it on the unit circle. At angles \(0^\circ\) and \(180^\circ\), which correspond to points on the right and left of the circle's center, the coordinates are \((1, 0)\) and \((-1, 0)\) respectively.

• At \(0^\circ\), \( \sin \theta = 0 \) and \( \cos \theta = 1 \), leading to \( \tan \theta = \frac{0}{1} = 0 \).
• At \(180^\circ\), \( \sin \theta = 0 \) and \( \cos \theta = -1 \), leading to \( \tan \theta = \frac{0}{-1} = 0 \).

This visual understanding supports the algebraic approach, allowing us to see why certain trigonometric values repeat as the angle goes around the circle. The period of the tangent function results in this regular, cyclic behavior observed in the unit circle.