Problem 55
Question
Solve the given equation. $$3 \tan \theta \sin \theta-2 \tan \theta=0$$
Step-by-Step Solution
Verified Answer
\(\theta = n\pi\) or \(\theta = \sin^{-1}\left(\frac{2}{3}\right) + 2n\pi\) or \(\theta = \pi - \sin^{-1}\left(\frac{2}{3}\right) + 2n\pi\), \(n\) is an integer.
1Step 1: Factor out common terms
The given equation is \(3 \tan \theta \sin \theta - 2 \tan \theta = 0\). Notice that \(\tan \theta\) is a common factor in both terms. Factor it out to get:\[\tan \theta (3 \sin \theta - 2) = 0\].
2Step 2: Set each factor to zero
To solve the equation \(\tan \theta (3 \sin \theta - 2) = 0\), set each factor equal to zero separately:1. \(\tan \theta = 0\)2. \(3 \sin \theta - 2 = 0\).
3Step 3: Solve \(\tan \theta = 0\)
The tangent of an angle is zero when the angle is an integer multiple of \(\pi\). Therefore, solve \(\tan \theta = 0\) for \(\theta\):\[\theta = n\pi, \text{ where } n \text{ is any integer.}\]
4Step 4: Solve \(3 \sin \theta - 2 = 0\)
Isolate \(\sin \theta\) by adding 2 to both sides and then dividing by 3:\[3 \sin \theta = 2 \\sin \theta = \frac{2}{3}\]Find \(\theta\) such that \(\sin \theta = \frac{2}{3}\). The solutions are:\[\theta = \sin^{-1}\left(\frac{2}{3}\right) + 2n\pi \text{ or } \theta = \pi - \sin^{-1}\left(\frac{2}{3}\right) + 2n\pi\] where \(n\) is any integer.
Key Concepts
FactoringTangent FunctionSine FunctionInverse Sine Function
Factoring
Factoring is a crucial mathematical process that simplifies expressions and equations. In the given equation, we have the expression \(3 \tan \theta \sin \theta - 2 \tan \theta = 0\). Here, \(\tan \theta\) appears as a common factor. Factoring involves extracting this common factor from each term. By doing so, we rewrite the equation as \(\tan \theta (3 \sin \theta - 2) = 0\).
Factoring is a method that can make it easier to solve equations because it reduces complexity. This approach breaks down a difficult equation into simpler parts, often leading to solutions more directly.
When encountering any polynomial or algebraic expression, always look for factors that repeat in each term. Simplifying expressions through factoring is not only helpful for trigonometric equations but is also a widely used technique throughout algebra.
Factoring is a method that can make it easier to solve equations because it reduces complexity. This approach breaks down a difficult equation into simpler parts, often leading to solutions more directly.
When encountering any polynomial or algebraic expression, always look for factors that repeat in each term. Simplifying expressions through factoring is not only helpful for trigonometric equations but is also a widely used technique throughout algebra.
- Identify common factors in each term.
- Extract the common factor.
- Rewrite the equation with the factorized expression.
Tangent Function
The tangent function, denoted \(\tan \theta\), is one of the basic trigonometric functions. It is defined as the ratio of the sine and cosine of an angle \(\theta\). Mathematically, this relationship is expressed as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
The tangent function has specific properties: it is undefined wherever \(\cos \theta = 0\); typically at \(\theta = \frac{\pi}{2} + n\pi\). The function itself is periodic, with a period of \(\pi\).
Understanding where \(\tan \theta = 0\) is essential for solving the equation. Tangent equals zero whenever \(\sin \theta = 0\), which happens at integer multiples of \(\pi\) (i.e., \(\theta = n\pi\), where \(n\) is an integer).
Keep in mind:
The tangent function has specific properties: it is undefined wherever \(\cos \theta = 0\); typically at \(\theta = \frac{\pi}{2} + n\pi\). The function itself is periodic, with a period of \(\pi\).
Understanding where \(\tan \theta = 0\) is essential for solving the equation. Tangent equals zero whenever \(\sin \theta = 0\), which happens at integer multiples of \(\pi\) (i.e., \(\theta = n\pi\), where \(n\) is an integer).
Keep in mind:
- \(\tan \theta\) is periodic with a period of \(\pi\).
- Tangent is zero at integer multiples of \(\pi\).
- It is undefined whenever \(\cos \theta = 0\).
Sine Function
The sine function, represented as \(\sin \theta\), is a trigonometric function that gives the ratio of the opposite side to the hypotenuse in a right-angled triangle. In the equation \(3 \sin \theta - 2 = 0\), solving for \(\sin \theta\) is necessary to find the possible angles \(\theta\).
Firstly, isolate \(\sin \theta\) by rearranging the equation: \[3 \sin \theta = 2 \quad \text{so} \quad \sin \theta = \frac{2}{3}\] The sine values range from -1 to 1, and any solution should be in this interval.
The sine function is also periodic, repeating every \(2\pi\). So, solutions for \(\sin \theta = \frac{2}{3}\) can include angles where their sine value equals \(\frac{2}{3}\) but have been rotated by \(2n\pi\), where \(n\) is an integer.
Firstly, isolate \(\sin \theta\) by rearranging the equation: \[3 \sin \theta = 2 \quad \text{so} \quad \sin \theta = \frac{2}{3}\] The sine values range from -1 to 1, and any solution should be in this interval.
The sine function is also periodic, repeating every \(2\pi\). So, solutions for \(\sin \theta = \frac{2}{3}\) can include angles where their sine value equals \(\frac{2}{3}\) but have been rotated by \(2n\pi\), where \(n\) is an integer.
- Sine is periodic with a period of \(2\pi\).
- Values are limited to the range -1 to 1.
Inverse Sine Function
The inverse sine function, denoted as \(\sin^{-1} x\) or \(\arcsin x\), is used to determine the angle whose sine is a given number. When solving \(\sin \theta = \frac{2}{3}\), we use the inverse sine function to find \(\theta\).
The expression \(\theta = \sin^{-1}(\frac{2}{3})\) provides the principal value, typically within the range of \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). However, in many trigonometric problems, we need to consider all possible angles that satisfy the equation.
For the sine function, solutions can also be found using the symmetry of the sine curve: \[\theta = \pi - \sin^{-1} \left(\frac{2}{3}\right) + 2n\pi\]This captures all angles within the periodic cycle that fulfill \(\sin \theta = \frac{2}{3}\).
Key points about inverse sine function:
The expression \(\theta = \sin^{-1}(\frac{2}{3})\) provides the principal value, typically within the range of \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). However, in many trigonometric problems, we need to consider all possible angles that satisfy the equation.
For the sine function, solutions can also be found using the symmetry of the sine curve: \[\theta = \pi - \sin^{-1} \left(\frac{2}{3}\right) + 2n\pi\]This captures all angles within the periodic cycle that fulfill \(\sin \theta = \frac{2}{3}\).
Key points about inverse sine function:
- Finds angles based on sine values.
- Principal values are generally between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
- Other solutions derived through symmetry: \(\theta = \pi - \sin^{-1}(x)\).
Other exercises in this chapter
Problem 54
Verify the identity. $$\sin ^{2} \alpha+\cos ^{2} \alpha+\tan ^{2} \alpha=\sec ^{2} \alpha$$
View solution Problem 55
Write the product as a sum. $$\sin 2 x \cos 3 x$$
View solution Problem 55
Solve the equation by first using a Sum-to-Product Formula. $$\cos 4 \theta+\cos 2 \theta=\cos \theta$$
View solution Problem 55
Verify the identity. $$\tan ^{2} \theta-\sin ^{2} \theta=\tan ^{2} \theta \sin ^{2} \theta$$
View solution