Solve: [tex]$\cos (x+\pi)=\frac{1}{2}$[/tex] over the interval [tex]$\left[\frac{\pi}{2}, \pi\right]$[/tex]
Answer (2)
Use the identity cos ( x + π ) = − cos ( x ) to rewrite the equation as − cos ( x ) = 2 1 , which simplifies to cos ( x ) = − 2 1 .
Find the general solutions for x such that cos ( x ) = − 2 1 . The general solutions are x = 3 2 π + 2 nπ and x = 3 4 π + 2 nπ , where n is an integer.
Determine which of the general solutions fall within the interval [ 2 π , π ] .
The only solution in the given interval is 3 2 π .
Explanation
Problem Analysis We are given the equation cos ( x + π ) = 2 1 and asked to solve for x in the interval [ 2 π , π ] .
Applying Trigonometric Identity Using the trigonometric identity cos ( x + π ) = − cos ( x ) , we can rewrite the equation as − cos ( x ) = 2 1 . Multiplying both sides by − 1 , we get cos ( x ) = − 2 1 .
Finding the Solution We need to find the values of x in the interval [ 2 π , π ] such that cos ( x ) = − 2 1 . We know that cos ( 3 2 π ) = − 2 1 . Since 2 π = 6 3 π and π = 6 6 π , we have 6 3 π < 6 4 π = 3 2 π < 6 6 π . Therefore, 3 2 π is in the interval [ 2 π , π ] .
General Solutions The general solutions for cos ( x ) = − 2 1 are x = 3 2 π + 2 nπ and x = 3 4 π + 2 nπ , where n is an integer. When n = 0 , we have x = 3 2 π and x = 3 4 π . Since we are looking for solutions in the interval [ 2 π , π ] , we check if these solutions fall within the interval. We already determined that 3 2 π is in the interval. However, 3 4 π is greater than π , so it is not in the interval. For any other integer value of n , the solutions will be outside the interval.
Final Answer Therefore, the only solution in the given interval is x = 3 2 π .
Examples
Trigonometric equations are used in physics to model oscillatory motion, such as the motion of a pendulum or the vibration of a string. They also appear in electrical engineering when analyzing alternating current circuits. Solving trigonometric equations within a specific interval is crucial for determining the state of a system at a particular time or within a certain range of conditions. For example, finding the angle of a pendulum at a given moment or the voltage in an AC circuit over a specific time period.
By using the identity cos ( x + π ) = − cos ( x ) , the equation simplifies to cos ( x ) = − 2 1 . The only solution within the interval [ 2 π , π ] is 3 2 π .
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