Which of the following is not equivalent to cos ((pi)/(5)) ? cos ((11 pi)/(5)) cos(-(9pi)/(5)) cos ((14 pi)/(5)) cos ((9pi)/(5))

Question

Which of the following is not equivalent to  cos ((pi)/(5)) ?  cos ((11 pi)/(5))  cos(-(9pi)/(5))  cos ((14 pi)/(5))  cos ((9pi)/(5))

Bytelearn · Accepted Answer

Properties of Cosine Function: Understand the properties of the cosine function. The cosine function has a period of 2π, which means that cos(θ) = cos(θ + 2πk) for any integer k. Also, cosine is an even function, so cos(θ) = cos(-θ).Evaluate cos ((11 pi)/(5)): Evaluate cos ((11 pi)/(5)). Using the periodic property of cosine, we can subtract 2π (which is the same as 10π/5) from (11π/5) to find an equivalent angle within the first period. cos ((11 pi)/(5)) = cos ((11 pi)/(5) - (10 pi)/(5)) = cos ((1 pi)/(5)) Since (1π/5) is equivalent to (π/5), this expression is equivalent to cos ((pi)/(5)).Evaluate cos (-(9pi)/(5)): Evaluate cos (-(9pi)/(5)). Using the even property of cosine, we know that cos(θ) = cos(-θ). cos (-(9pi)/(5)) = cos ((9pi)/(5)) Now, we can add 2π (which is the same as 10π/5) to (9π/5) to find an equivalent angle within the first period. cos ((9pi)/(5)) = cos ((9pi)/(5) + (10 pi)/(5)) = cos ((19 pi)/(5)) We can subtract 2π again to find an equivalent angle. cos ((19 pi)/(5)) = cos ((19 pi)/(5) - (10 pi)/(5)) = cos ((9 pi)/(5)) Since we have already established that cos (-(9pi)/(5)) = cos ((9pi)/(5)), this expression is equivalent to cos ((pi)/(5)).Evaluate cos ((14 pi)/(5)): Evaluate cos ((14 pi)/(5)). Using the periodic property of cosine, we can subtract 2π (which is the same as 10π/5) from (14π/5) to find an equivalent angle within the first period. cos ((14 pi)/(5)) = cos ((14 pi)/(5) - (10 pi)/(5)) = cos ((4 pi)/(5)) Since (4π/5) is not equivalent to (π/5), this expression is not equivalent to cos ((pi)/(5)).Evaluate cos ((9pi)/(5)): Evaluate cos ((9pi)/(5)). Using the periodic property of cosine, we can subtract 2π (which is the same as 10π/5) from (9π/5) to find an equivalent angle within the first period. cos ((9pi)/(5)) = cos ((9pi)/(5) - (10 pi)/(5)) = cos (-(1 pi)/(5)) Using the even property of cosine, we know that cos(θ) = cos(-θ). cos (-(1 pi)/(5)) = cos ((1 pi)/(5)) Since (1π/5) is equivalent to (π/5), this expression is equivalent to cos ((pi)/(5)).

Which of the following is not equivalent to $\cos \frac{\pi}{5}$ ? $\newline$ $\cos \frac{11 \pi}{5}$ $\newline$ $\cos \left(-\frac{9 \pi}{5}\right)$ $\newline$ $\cos \frac{14 \pi}{5}$ $\newline$ $\cos \frac{9 \pi}{5}$

Full solution

More problems from Multiplication with rational exponents

Related topics