Problem 91 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The equation \(1 - \cos \theta = \sin \theta\) is not an identity because it does not hold for all values of \(\theta\). One such value is \(\theta = 0\).

Step 1: Understanding the trigonometric relationship

The trigonometric relationship says that \( \sin^2 \theta + \cos^2 \theta = 1 \). But this doesn't imply that \(1 - \cos \theta = \sin \theta \) because there's no guarantee that the square root of \(1 - \cos \theta\) will be \(\sin \theta\). Squaring both sides might lose information about the original equation.

Step 2: Testing a value

One way to test whether the equation is an identity is to substitute a known value of \(\theta\), say \( \theta = 0\). On substituting, we get \(1-\cos 0 = \sin 0 \) which simplifies to \(1-1 \neq 0\). Hence the equation \(1 - \cos \theta = \sin \theta\) is not an identity and \(\theta = 0\) is a value for which the equation does not hold true.