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Show that \(\left(\sin(a)\right)^7 + \left(\cos ( a)\right)^7< 1\) if \(0 < a < \pi/2\)

 Jun 27, 2023
 #4
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Given: 0 < a < pi/2

To prove: sin(a)^7 + cos(a)^7 < 1

Proof:

Since 0 < a < pi/2, sin(a) and cos(a) are both positive numbers less than 1.

For any positive number x less than 1, x^7 is also less than 1.

Therefore, sin(a)^7 and cos(a)^7 are both less than 1.

Adding two numbers less than 1 gives a number less than 2.

Therefore, sin(a)^7 + cos(a)^7 < 1.

Q.E.D.

Here is an explanation of the proof:

The first step is to show that sin(a) and cos(a) are both positive numbers less than 1. This is true because 0 < a < pi/2, which means that a is in the first quadrant. In the first quadrant, all trigonometric functions are positive.

The second step is to show that x^7 is less than 1 for any positive number x less than 1. This is true because as x gets closer to 0, x^7 gets closer to 0. Therefore, for any positive number x less than 1, x^7 is less than 1.

The third step is to add sin(a)^7 and cos(a)^7. Since both sin(a)^7 and cos(a)^7 are less than 1, their sum is also less than 1.

The fourth step is to conclude that sin(a)^7 + cos(a)^7 < 1. This is true because the sum of two numbers less than 1 is less than 2.

I hope this explanation is helpful. Please let me know if you have any other questions.

 Jun 27, 2023

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