Explain the mistake that is made. Solve 3 sin(2 x)=2 cosx on 0^∘ ≤θ≤180^∘. Solution: Use the dou

Name: Problem 121, Chapter 7: Algebra and Trigonometry
Uploaded: 2021-06-24T22:11:08-08:00
Duration: 33 s
Channel: James Kiss

step 1 In examining the worked -out problem, they should not divide by cosine of x. Instead, it must be

Question

Explain the mistake that is made. Solve $3 \sin (2 x)=2 \cos x$ on $0^{\circ} \leq \theta \leq 180^{\circ}$. Solution: Use the double-angle identity for the sine function. $3 \frac{3 \sin 2 x}{2 \sin x \cos x}=2 \cos x$ Simplify. $6 \sin x \cos x=2 \cos x$ Divide by $2 \cos x$. $3 \sin x=1$ Divide by $3 .$ $\sin x=\frac{1}{3}$ $\begin{array}{l}\text { Write the equivalent } \\ \text { inverse notation. }\end{array} \quad x=\sin ^{-1}\left(\frac{1}{3}\right)$ Use a calculator to approximate the solution. $x \approx 19.47^{\circ},$ QI solution The QII solution is: $x \approx 180^{\circ}-19.47^{\circ} \approx 160.53^{\circ}$ This is incorrect. What mistake was made?

   Explain the mistake that is made.   
Solve $3 \sin (2 x)=2 \cos x$ on $0^{\circ} \leq \theta \leq 180^{\circ}$.
Solution:
Use the double-angle identity for the sine function. $3 \frac{3 \sin 2 x}{2 \sin x \cos x}=2 \cos x$
Simplify. $6 \sin x \cos x=2 \cos x$
Divide by $2 \cos x$. $3 \sin x=1$
Divide by $3 .$ $\sin x=\frac{1}{3}$
$\begin{array}{l}\text { Write the equivalent } \\ \text { inverse notation. }\end{array} \quad x=\sin ^{-1}\left(\frac{1}{3}\right)$
Use a calculator to approximate the solution. $x \approx 19.47^{\circ},$ QI solution
The QII solution is:
$x \approx 180^{\circ}-19.47^{\circ} \approx 160.53^{\circ}$
This is incorrect. What mistake was made?

Algebra and Trigonometry

Cynthia Y. Young 4th Edition

Chapter 7, Problem 121

Instant Answer

Solved by Expert James Kiss

06/24/2021

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Explain the mistake that is made. Solve $3 \sin (2 x)=2 \cos x$ on $0^{\circ} \leq \theta \leq 180^{\circ}$. Solution: Use the double-angle identity for the sine function. $3 \frac{3 \sin 2 x}{2 \sin x \cos x}=2 \cos x$ Simplify. $6 \sin x \cos x=2 \cos x$ Divide by $2 \cos x$. $3 \sin x=1$ Divide by $3 .$ $\sin x=\frac{1}{3}$ $\begin{array}{l}\text { Write the equivalent } \\ \text { inverse notation. }\end{array} \quad x=\sin ^{-1}\left(\frac{1}{3}\right)$ Use a calculator to approximate the solution. $x \approx 19.47^{\circ},$ QI solution The QII solution is: $x \approx 180^{\circ}-19.47^{\circ} \approx 160.53^{\circ}$ This is incorrect. What mistake was made?

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Key Concepts

Double-Angle Identity

The double-angle identity for sine, sin(2x) = 2 sin(x) cos(x), is a fundamental trigonometric formula used to simplify expressions involving sine functions of double angles. It allows the transformation of the equation into a product form, making it easier to solve by identifying common factors.

Division by a Variable Expression

When an equation is simplified by dividing by an expression that involves a variable, it is critical to ensure that the divisor is not zero. Dividing by an expression that could be zero might remove valid solutions from the set since the equation would be undefined at those points. This is a common pitfall in solving equations and must be carefully checked to avoid discarding potential solutions.

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