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Chapter 6: Problem 2

Complete each Pythagorean identity. $$\sin ^{2} x=1-$$

Short Answer

Expert verified
The completed Pythagorean identity is \(\sin^{2} x = 1 - \cos^{2} x\)

Step by step solution

01

Recall the Pythagorean Identity

The Pythagorean identity states that for any angle \(x\), the sine squared of that angle plus the cosine squared of that angle is equal to one. This can be written as \(\sin^{2} x + \cos^{2} x = 1\)
02

Solve for Sine Squared

In order to complete the equation asked in the problem, it will be necessary to isolate the sine squared term. This can be done by subtracting \(\cos^{2} x\) from both sides of the equation. The resulting equation is \(\sin^{2} x = 1 - \cos^{2} x\)
03

Insert into the given identity

Now, the right only part of the identity derived in the previous step can be replaced in the original given incomplete identity, getting the complete equation: \(\sin^{2} x = 1 - \cos^{2} x\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Trigonometric Identities
Trigonometric identities are equations that are true for all values of the variables they contain. They play a crucial role in simplifying trigonometric expressions and solving trigonometric equations. One of the fundamental sets of identities you'll encounter in trigonometry is the Pythagorean identities. These are based on the Pythagorean theorem, which relates the sides of a right triangle.

There are three Pythagorean identities in trigonometry, but the one most pertinent to our exercise is the following: \[\sin^2x + \cos^2x = 1\] This relates the squares of the sine and cosine of an angle to the number 1. It's a powerful tool for transformation and evaluation of trigonometric expressions since it connects two influential trigonometric functions, sine and cosine, paving the way for simplification and problem-solving.
Diving Deep into 'sin squared x'
The term \(\sin^2 x\), which you might see written as 'sine squared of x', refers to the square of the sine of an angle x. The sine function measures the ratio of the opposite side to the hypotenuse in a right-angled triangle. When you square this ratio, you're essentially multiplying the sine of an angle by itself.

Understanding \(\sin^2 x\) is fundamental because it frequently appears in trigonometric equations and identities, including the Pythagorean identity we're exploring. It's also useful in calculating the areas of certain shapes and in analyzing the components of physical phenomena, such as waves.
Focusing on 'cos squared x'
Similarly, \(\cos^2 x\) is the abbreviation for 'cosine squared of x', the square of the cosine function for an angle x. Cosine describes the ratio of the adjacent side to the hypotenuse in a right-angled triangle. When this ratio is squared, it's acting on itself in the same way as the sine function when squared.

\(\cos^2 x\) regularly comes up not only in trigonometric identities like the Pythagorean identity but also in calculus, such as when finding the derivative or integral of sine functions. \(\cos^2 x\) and \(\sin^2 x\) are intimately linked in trigonometry; they complement each other in a way that the total of their squares always equals 1, a recurrent and essential trait used to unravel more complex trigonometric problems.

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