Problem 21 - Algebra 2

Question

Solve each equation for all values of \(\theta\) if \(\theta\) is measured in degrees. \(\tan \theta=\sin \theta\)

Step-by-Step Solution

Answer

\(\theta = n \times 180^\circ\), where \(n\) is an integer.

Step 1: Recall the Trigonometric Identities

First, remember the basic trigonometric identities: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). We will use this identity to rewrite the given equation.

Step 2: Substitute and Simplify the Equation

Substitute \(\tan \theta\) with \(\frac{\sin \theta}{\cos \theta}\) in the equation. This gives us \(\frac{\sin \theta}{\cos \theta} = \sin \theta\). Multiply both sides by \(\cos \theta\) to eliminate the fraction, resulting in \(\sin \theta = \sin \theta \cos \theta\).

Step 3: Factor Out Common Terms

Rewrite the equation from the previous step: \(\sin \theta - \sin \theta \cos \theta = 0\). Factor out \(\sin \theta\), leading to \(\sin \theta (1 - \cos \theta) = 0\). This provides two possible solutions: \(\sin \theta = 0\) and \(1 - \cos \theta = 0\).

Step 4: Solve for \(\theta\) from \(\sin \theta = 0\)

The equation \(\sin \theta = 0\) can be solved by recognizing that sine is zero at integer multiples of 180 degrees. Thus, \(\theta = n \times 180^\circ\), where \(n\) is an integer.

Step 5: Solve for \(\theta\) from \(1 - \cos \theta = 0\)

Simplify \(1 - \cos \theta = 0\) to find \(\cos \theta = 1\). Cosine is 1 at \(\theta = 0^\circ + 360^\circ m\), where \(m\) is an integer. Thus, \(\theta = m \times 360^\circ\).

Step 6: Combine Solutions

Combine the solutions from both cases. Since \(\{0^\circ, 180^\circ, 360^\circ, \ldots\}\) are special cases of multiples of 180, the general solution can be written as \(\theta = n \times 180^\circ\), accommodating all solutions.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools that help simplify and solve equations involving trigonometric functions. They are mathematical truths based on the properties of the sine, cosine, and tangent functions. Here are some fundamental identities that are frequently used:

• Pythagorean Identities: \( ext{sin}^2 \theta + ext{cos}^2 \theta = 1\)
• Quotient Identities: \( an \theta = \frac{\sin \theta}{\cos \theta}\)
• Reciprocal Identities: \( ext{sec}\theta = \frac{1}{\cos \theta}\), \( ext{csc}\theta = \frac{1}{\sin \theta}\)

In this exercise, we use the quotient identity for tangent. This identity helps us rewrite the equation \( \tan \theta = \sin \theta \) into an equation that is easier to manipulate. By expressing \( \tan \theta \) as \( \frac{\sin \theta}{\cos \theta} \), we can work towards eliminating the fraction and simplifying the equation further.

Solving Equations

Solving trigonometric equations involves finding the angles that satisfy the given conditions. It usually involves manipulating the equation to simplify it down to more basic trigonometric expressions.In our problem, after using the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), the equation becomes \(\frac{\sin \theta}{\cos \theta} = \sin \theta\). By multiplying both sides by \(\cos \theta\), we eliminate the fraction:\[\sin \theta = \sin \theta \cos \theta\]The next step is to factor out common terms to simplify the equation further:\[\sin \theta (1 - \cos \theta) = 0\]This gives us two separate equations:

• \(\sin \theta = 0\)
• \(1 - \cos \theta = 0\)

Each equation allows us to solve for \(\theta\), which we can do by recognizing the specific values of angles where sine equals zero and cosine equals one.

Angle Measurement

Angle measurement is crucial when solving trigonometric equations. Typically, angles can be measured in degrees or radians. In this exercise, we work with degrees, which are a more familiar unit for most people.Angles in degrees can be thought of as fractions of a full circle, where a full circle is \(360^\circ\). Specific angles have well-known trigonometric values:

• \(\sin \theta = 0\): This occurs at angles like \(0^\circ, 180^\circ, 360^\circ\).
• \(\cos \theta = 1\): This occurs at angles like \(0^\circ, 360^\circ\).

When solving for values of \(\theta\), we use these known values to express the solution in a general form. For example, \(\theta = n \times 180^\circ\) or \(\theta = m \times 360^\circ\), where \(n\) and \(m\) are integers, represent all angles where the given conditions hold true. This approach helps us capture all the possible solutions by recognizing patterns in trigonometric functions.