Which of the following best explains why [tex]$\cos \frac{2 \pi}{3} \neq \cos \frac{5 \pi}{3}$[/tex] ?
A. The angles do not have the same reference angle.
B. Cosine is negative in the second quadrant and positive in the fourth quadrant.
C. Cosine is positive in the second quadrant and negative in the fourth quadrant.
D. The angles do not have the same reference angle or the same sign.
Answer (2)
Determine the reference angles for 3 2 π and 3 5 π , which are both 3 π .
Identify that 3 2 π is in the second quadrant, where cosine is negative.
Identify that 3 5 π is in the fourth quadrant, where cosine is positive.
Conclude that cos ( 3 2 π ) = cos ( 3 5 π ) because cosine is negative in the second quadrant and positive in the fourth quadrant. $\boxed{\text{Cosine is negative in the second quadrant and positive in the fourth quadrant.}}
Explanation
Problem Analysis Let's analyze the given trigonometric problem. We are asked to determine the best explanation for why cos ( 3 2 π ) = cos ( 3 5 π ) . We will analyze the reference angles and quadrants of the given angles.
Find Reference Angles First, let's find the reference angles for both 3 2 π and 3 5 π .
The angle 3 2 π is in the second quadrant. Its reference angle is π − 3 2 π = 3 π .
The angle 3 5 π is in the fourth quadrant. Its reference angle is 2 π − 3 5 π = 3 π .
So, both angles have the same reference angle, which is 3 π .
Determine Quadrants and Signs Now, let's determine the quadrants in which the angles lie and the signs of the cosine function in those quadrants.
The angle 3 2 π is in the second quadrant, where cosine is negative. Therefore, cos ( 3 2 π ) = − cos ( 3 π ) = − 2 1 .
The angle 3 5 π is in the fourth quadrant, where cosine is positive. Therefore, cos ( 3 5 π ) = cos ( 3 π ) = 2 1 .
Explain the Difference Since the reference angles are the same, but the signs of the cosine function in the respective quadrants are different, this explains why cos ( 3 2 π ) = cos ( 3 5 π ) .
Final Answer Based on our analysis, the best explanation is that cosine is negative in the second quadrant and positive in the fourth quadrant.
Examples
Understanding the cosine function in different quadrants is crucial in fields like electrical engineering, where alternating current (AC) waveforms are modeled using sinusoidal functions. For example, when analyzing AC circuits, knowing the phase relationship between voltage and current, which is determined by the cosine of the phase angle, helps engineers design efficient power systems and prevent equipment damage. If the phase angle is 3 2 π , the power factor is negative, indicating reactive power, while a phase angle of 3 5 π gives a positive power factor, indicating active power. This distinction is vital for optimizing energy usage and ensuring stable grid operation.
The reason cos 3 2 π = cos 3 5 π is that they share the same reference angle, 3 π , but in the second quadrant, cosine is negative, while in the fourth quadrant, cosine is positive. Thus, the answer is B: Cosine is negative in the second quadrant and positive in the fourth quadrant.
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