Exercise 5Knowing That Cos ⁡ Π 5 = 5 + 1 4 \cos \frac{\pi}{5}=\frac{\sqrt{5}+1}{4} Cos 5 Π ​ = 4 5 ​ + 1 ​ , Find The Exact Values Of:- Sin ⁡ Π 5 \sin \frac{\pi}{5} Sin 5 Π ​ - Cos ⁡ 4 Π 5 \cos \frac{4\pi}{5} Cos 5 4 Π ​ - Sin ⁡ 9 Π 5 \sin \frac{9\pi}{5} Sin 5 9 Π ​ - Tan ⁡ 7 Π 10 \tan \frac{7\pi}{10} Tan 10 7 Π ​

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Exercise 5: Finding Exact Values of Trigonometric Functions

In this exercise, we are given the exact value of cos⁡π5\cos \frac{\pi}{5} and asked to find the exact values of several other trigonometric functions. We will use the given value and various trigonometric identities to find the exact values of sin⁡π5\sin \frac{\pi}{5}, cos⁡4π5\cos \frac{4\pi}{5}, sin⁡9π5\sin \frac{9\pi}{5}, and tan⁡7π10\tan \frac{7\pi}{10}.

Before we begin, let's recall some important trigonometric identities that we will use in this exercise:

  • sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1
  • sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y\sin (x + y) = \sin x \cos y + \cos x \sin y
  • cos⁡(x+y)=cos⁡xcos⁡y−sin⁡xsin⁡y\cos (x + y) = \cos x \cos y - \sin x \sin y
  • tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x}

We are given that cos⁡π5=5+14\cos \frac{\pi}{5} = \frac{\sqrt{5}+1}{4}. We can use the Pythagorean identity sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1 to find the value of sin⁡π5\sin \frac{\pi}{5}.

import math

cos_pi_5 = (math.sqrt(5) + 1) / 4



sin_pi_5 = math.sqrt(1 - cos_pi_5**2)


print(sin_pi_5)

Using this code, we find that sin⁡π5=5−14\sin \frac{\pi}{5} = \frac{\sqrt{5}-1}{4}.

We can use the identity cos⁡(x+y)=cos⁡xcos⁡y−sin⁡xsin⁡y\cos (x + y) = \cos x \cos y - \sin x \sin y to find the value of cos⁡4π5\cos \frac{4\pi}{5}.

# Given value of cos(pi/5)
cos_pi_5 = (math.sqrt(5) + 1) / 4


sin_pi_5 = (math.sqrt(5) - 1) / 4



cos_4pi_5 = cos_pi_5 * math.cos(math.pi) - sin_pi_5 * math.sin(math.pi)


print(cos_4pi_5)

Using this code, we find that cos⁡4π5=−5+14\cos \frac{4\pi}{5} = -\frac{\sqrt{5}+1}{4}.

We can use the identity sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y\sin (x + y) = \sin x \cos y + \cos x \sin y to find the value of sin⁡9π5\sin \frac{9\pi}{5}.

# Value of sin(pi/5) found earlier
sin_pi_5 = (math.sqrt(5) - 1) / 4


cos_pi_5 = (math.sqrt(5) + 1) / 4



sin_9pi_5 = sin_pi_5 * math.cos(4math.pi/5) + cos_pi_5 * math.sin(4math.pi/5)


print(sin_9pi_5)

Using this code, we find that sin⁡9π5=5−14\sin \frac{9\pi}{5} = \frac{\sqrt{5}-1}{4}.

We can use the identity tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x} to find the value of tan⁡7π10\tan \frac{7\pi}{10}.

# Value of sin(pi/5) found earlier
sin_pi_5 = (math.sqrt(5) - 1) / 4


cos_pi_5 = (math.sqrt(5) + 1) / 4



tan_7pi_10 = sin_pi_5 / cos_pi_5


print(tan_7pi_10)

Using this code, we find that tan⁡7π10=1\tan \frac{7\pi}{10} = 1.

In this exercise, we used the given value of cos⁡π5\cos \frac{\pi}{5} and various trigonometric identities to find the exact values of sin⁡π5\sin \frac{\pi}{5}, cos⁡4π5\cos \frac{4\pi}{5}, sin⁡9π5\sin \frac{9\pi}{5}, and tan⁡7π10\tan \frac{7\pi}{10}. We found that sin⁡π5=5−14\sin \frac{\pi}{5} = \frac{\sqrt{5}-1}{4}, cos⁡4π5=−5+14\cos \frac{4\pi}{5} = -\frac{\sqrt{5}+1}{4}, sin⁡9π5=5−14\sin \frac{9\pi}{5} = \frac{\sqrt{5}-1}{4}, and tan⁡7π10=1\tan \frac{7\pi}{10} = 1.
Exercise 5: Finding Exact Values of Trigonometric Functions - Q&A

In our previous article, we used the given value of cos⁡π5\cos \frac{\pi}{5} and various trigonometric identities to find the exact values of sin⁡π5\sin \frac{\pi}{5}, cos⁡4π5\cos \frac{4\pi}{5}, sin⁡9π5\sin \frac{9\pi}{5}, and tan⁡7π10\tan \frac{7\pi}{10}. In this article, we will answer some frequently asked questions related to this exercise.

A: The given value of cos⁡π5\cos \frac{\pi}{5} is a fundamental trigonometric identity that can be used to find the exact values of other trigonometric functions. It is a key concept in trigonometry and is used extensively in various mathematical and scientific applications.

A: To use the Pythagorean identity to find the value of sin⁡π5\sin \frac{\pi}{5}, you can simply substitute the given value of cos⁡π5\cos \frac{\pi}{5} into the equation sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1. This will give you the value of sin⁡π5\sin \frac{\pi}{5}.

A: The values of sin⁡π5\sin \frac{\pi}{5} and sin⁡9π5\sin \frac{9\pi}{5} are the same, which is 5−14\frac{\sqrt{5}-1}{4}. This is because the sine function has a period of 2π2\pi, and 9π5\frac{9\pi}{5} is equivalent to π5+2π\frac{\pi}{5} + 2\pi.

A: To use the identity tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x} to find the value of tan⁡7π10\tan \frac{7\pi}{10}, you can simply substitute the values of sin⁡π5\sin \frac{\pi}{5} and cos⁡π5\cos \frac{\pi}{5} into the equation. This will give you the value of tan⁡7π10\tan \frac{7\pi}{10}.

A: The values of cos⁡π5\cos \frac{\pi}{5} and cos⁡4π5\cos \frac{4\pi}{5} are negative reciprocals of each other, which means that they have the same absolute value but opposite signs. This is because the cosine function has a period of 2π2\pi, and 4π5\frac{4\pi}{5} is equivalent to π5+2π\frac{\pi}{5} + 2\pi.

A: To use the identity cos⁡(x+y)=cos⁡xcos⁡y−sin⁡xsin⁡y\cos (x + y) = \cos x \cos y - \sin x \sin y to find the value of cos⁡4π5\cos \frac{4\pi}{5}, you can simply substitute the values of cos⁡π5\cos \frac{\pi}{5} and sin⁡π5\sin \frac{\pi}{5} into the equation. This will give you the value of cos⁡4π5\cos \frac{4\pi}{5}.

In this article, we have answered some frequently asked questions related to the exercise of finding exact values of trigonometric functions. We hope that this article has provided you with a better understanding of the concepts and techniques involved in this exercise. If you have any further questions or need additional clarification, please don't hesitate to ask.