Problem 100 - Algebra and Trigonometry

Question

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$ \sin 2 x+\cos x=0 $$

Step-by-Step Solution

Answer

There are four solutions to the given equation which are \( x = \pi/2, 3\pi/2, 7\pi/6, 11\pi/6 \).

Step 1: Apply Double-Angle Identity

Use the double-angle identity for sine to rewrite the equation in terms of \(x\). Replace \( \sin(2x) \) by \( 2 \sin(x) \cos(x) \) in the equation, yielding \( 2 \sin(x) \cos(x) + \cos(x) = 0 \).

Step 2: Factor The Equation

Factor the above equation by taking \( \cos(x) \) common to get: \( \cos(x) (2 \sin(x) + 1) = 0 \).

Step 3: Set Each Factor Equal to Zero

Set each factor equal to zero to find the value of \( x \): \( \cos(x) = 0 \) and \( 2 \sin(x)+1 =0 \).

Step 4: Solve the Two Equations

From \( \cos(x) = 0\), we get \( x = \pi/2, 3\pi/2 \).From \( 2 \sin(x)+1 = 0 \), we get \( \sin(x) = -1/2 \), which gives \( x = 7\pi/6, 11\pi/6 \).

Step 5: Check the Validity

Since all solutions are within the range \( [0, 2\pi) \), all results are valid.

Key Concepts

Double-Angle Identity

The double-angle identity is a critical tool in trigonometry that simplifies expressions involving angles twice a given angle. In this exercise, the identity used is specifically for sine. The double-angle identities for sine is given by:\[\sin(2x) = 2\sin(x)\cos(x)\]This identity helps to transform the original equation \(\sin(2x) + \cos(x) = 0\) into a more manageable form. By applying the identity, it becomes \(2\sin(x)\cos(x) + \cos(x) = 0\). The purpose of using such identities is to rewrite functions involving multiple angles in terms of single angles. Understanding and applying the double-angle identity makes solving trigonometric equations more straightforward. This is particularly useful when equations involve products of trigonometric functions or need simplification for easier analysis.

Factoring Trigonometric Expressions

Factoring is a mathematical technique used to simplify equations and solve for unknown values. When dealing with trigonometric equations, factoring plays a crucial role especially when the equation involves a common trigonometric factor.In the given step-by-step solution, factoring is used to break down the equation \(2\sin(x)\cos(x) + \cos(x) = 0\). Here, the expression is factored by taking \(\cos(x)\) as a common factor:\[\cos(x)(2\sin(x) + 1) = 0\] Once the expressions are factored, it becomes easier to solve because each factor can be set to zero independently:

• \(\cos(x) = 0\)
• \(2\sin(x) + 1 = 0\)

These simplified equations can then be solved with basic trigonometric properties. Factoring transforms complex equations into simpler components, making it much more efficient to find solutions.

Unit Circle

Understanding the unit circle is essential for solving trigonometric equations, particularly those that result from factoring and other transformations. The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane.Key points along the unit circle correspond to important angle measures in radians and help determine the sine and cosine values of these angles. For example:

• At \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\), \(\cos(x) = 0\)
• At \(\frac{7\pi}{6}\) and \(\frac{11\pi}{6}\), \(\sin(x) = -\frac{1}{2}\)

These angles represent the solutions to our factored equations from earlier steps. The unit circle is invaluable because it provides exact values for trigonometric functions at various key angles, without needing complicated calculations. Understanding the unit circle can therefore aid tremendously in solving trigonometric equations efficiently and accurately.