Problem 51 - Algebra 2

Question

Find the complete solution in radians of each equation. $$ \tan \theta \sin \theta=3 \sin \theta $$

Step-by-Step Solution

Answer

The solutions to the given equation are \( \theta = n\pi \) where \( n \) is an integer, and \( \theta = \tan^{-1} 3 + n\pi \) where \( n \) is an integer.

Step 1: Simplify the given equation

Firstly, observe the equation \( \tan \theta \sin \theta = 3 \sin \theta \). Since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), the equation can be rewritten as \( \frac{\sin^2 \theta}{\cos \theta} = 3 \sin \theta \). Next, multiply all terms by \( \cos \theta \) to clear the fraction which gives the new equation: \( \sin^2 \theta = 3 \sin \theta \cos \theta \).

Step 2: Rewrite in terms of \(\sin \theta\)

Subtract \(3 \sin \theta \cos \theta \) from both sides to obtain: \( \sin^2 \theta - 3 \sin \theta \cos \theta = 0\). It's possible to simplify this equation by factoring out \( \sin \theta \), leaving us with \( \sin \theta (\sin \theta - 3 \cos \theta) = 0\).

Step 3: Solve for \(\theta\)

Now, solve the equation by setting each factor equal to zero, and solve for the value of \( \theta \) in each case. So, \( \sin \theta = 0\) and \(\sin \theta - 3 \cos \theta = 0\). Solving for \( \theta \) in each case will give all the possible solutions.

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

The solution to \( \sin \theta = 0 \) is \( \theta = n\pi \) where \( n \) is an integer.

Step 5: Solve \( \sin \theta - 3 \cos \theta = 0 \)

The solution to \( \sin \theta - 3 \cos \theta = 0 \) can be rewritten as \( \tan \theta = 3 \) using the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta}\). Solving for \( \theta \), we have \( \theta = \tan^{-1} 3 + n\pi \) where \( n \) is an integer.

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan \theta \), is one of the primary trigonometric functions. It relates the angle \( \theta \) in a right triangle to the ratio of the side lengths. Specifically, it is the ratio of the opposite side to the adjacent side in a right triangle.

In terms of sine and cosine functions, the tangent function is expressed as:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

This identity is extremely useful because it allows us to rewrite equations involving tangent in terms of sine and cosine, which can be easier to work with.

In the exercise, we used this identity to simplify the equation \( \tan \theta \sin \theta = 3 \sin \theta \) to \( \frac{\sin^2 \theta}{\cos \theta} = 3 \sin \theta \). Simplifying equations often helps in solving them efficiently.

Sine Function

The sine function, denoted as \( \sin \theta \), is a fundamental trigonometric function. It provides us with the ratio of the length of the opposite side to the hypotenuse in a right triangle.

It's a periodic function, which means it repeats its values in regular intervals. The sine function has various identities associated with it, which are quite helpful in solving trigonometric equations.

In the problem presented, \( \sin \theta \) appears in both parts of the equation, \( \tan \theta \sin \theta = 3 \sin \theta \). By factoring \( \sin \theta \) out, we were able to transform the equation into \( \sin \theta (\sin \theta - 3 \cos \theta) = 0 \). This establishes two separate equations to solve: \( \sin \theta = 0 \) and \( \sin \theta - 3 \cos \theta = 0 \), facilitating the problem-solving process.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. They are critical tools in simplifying and solving trigonometric equations.

Some essential identities include:

• Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Tangent identity: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
• Angle sum identities

In this exercise, we primarily worked with the tangent identity, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Such identities help us rewrite trigonometric equations in forms that are easier to solve. By substituting these identities, we can often simplify factors or eliminate certain terms, making it possible to solve equations step by step.

Radians

Radians are a way of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians align the angle measurement with the arc length. In a circle, an angle of one radian results when the arc length is equal to the radius of the circle.

In terms of conversions, the entire circle is \( 2\pi \) radians because the circumference of a circle is \( 2\pi r \), where \( r \) is the radius. Thus:

• \( 180^\circ = \pi \) radians
• \( 1 \text{ radian} \approx 57.2958^\circ \)

The solution in the exercise used radians for presenting the complete solutions. When \( \sin \theta = 0 \), the solutions are at integer multiples of \( \pi \), represented as \( \theta = n\pi \) for integer \( n \). This reflects the periodic nature of the sine function in the radian measure.