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Chapter 5: Problem 75

Explain why the equation is not an identity and find one value of the variable for which the equation is not true. $$1-\cos \theta=\sin \theta$$

Short Answer

Expert verified
The equation \(1-\cos \theta=\sin \theta\) is not an identity. This is demonstrated by the value \( \theta = \pi \), for which the equation does not hold true.

Step by step solution

01

Understanding the given equation

We have the equation \(1-\cos \theta=\sin \theta\). If this equation is an identity, it will be true for every value of the variable \( \theta \). Our job is to prove that this equation is not an identity by finding a value of \( \theta \) that doesn't satisfy the equation.
02

Choosing a value to test

Typically, we start by testing simple values. It's often best to start with \( \theta = 0 \) or \( \theta = \pi / 2 \) because these are the points at which the sine and cosine functions take their maxima and minima. Let's start by substituting \( \theta = 0 \).
03

Substituting the value into the equation

By substituting \( \theta = 0 \) into our equation, we will find if this value satisfies it or not. On the left side of the equation, we get \(1-\cos(0)\), which simplifies to 0 since \(cos(0) = 1\). On the right-hand side, we have \(sin(0)\), which equals 0.
04

Conclusion

As both sides of the equation equal 0 when \( \theta = 0 \), it appears that this value of \( \theta \) satisfies the equation. This doesn't prove that it's an identity, though. Let's try another value, \( \theta = \pi / 2 \). Substituting this into the equation, on the left side we get \(1-\cos(\pi/2) = 1\), and on the right hand side we get \(sin(\pi/2) = 1\).
05

Concluding the analysis

Since both sides of the equation are equal when \( \theta = \pi / 2 \), this solution also satisfies the equation. However, this does not necessarily mean that the equation is an identity. An identity is only claimed if the equation holds true for all possible values of \( \theta \), not just for a few values we have tested. Let's test another value, say \( \theta = \pi \). Substituting this into the equation, on the left side we get \(1-\cos(\pi) = 2\), and on the right hand side we get \(sin(\pi) = 0\). Therefore, the equation does not hold true with \( \theta = \pi \), showing that the equation is not an identity.

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