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Chapter 5: Problem 82

Show that $$ \sin (\pi-\theta)=\sin \theta $$ for every angle \(\theta\).

Short Answer

Expert verified
Using the sine difference formula, we have \(\sin(\pi - \theta) = \sin\pi \cos\theta - \cos\pi \sin\theta\). Knowing that \(\sin\pi = 0\) and \(\cos\pi = -1\), we get \(\sin(\pi - \theta) = 0 - (-\sin\theta)\), which simplifies to \(\sin(\pi - \theta) = \sin\theta\). Thus, the given identity is proven.

Step by step solution

01

Recall the sine difference formula.

The sine difference formula is given by: \[ \sin(A - B) = \sin A \cos B - \cos A \sin B \] We will use this formula to prove the given formula, with A = 蟺 and B = 胃.
02

Apply the sine difference formula with A = 蟺 and B = 胃.

Using the sine difference formula, let A = 蟺 and B = 胃, we have: \[ \sin(\pi - \theta) = \sin\pi \cos\theta - \cos\pi \sin\theta \]
03

Evaluate the sine and cosine values for 胃.

Recall the values of sine and cosine at the angle 蟺 (180 degrees): \[ \sin\pi = 0 \] and \[ \cos\pi = -1 \]
04

Replace the sine and cosine values in the formula.

Replace the sine and cosine values in the sine difference formula with the results from step 3: \[ \sin(\pi - \theta) = (0)\cos\theta - (-1)\sin\theta \]
05

Simplify the expression.

Simplify the expression to get the final result: \[ \sin(\pi - \theta) = 0 - (-\sin\theta) \] This simplifies to: \[ \sin(\pi - \theta) = \sin\theta \] The proof is now complete. We have shown that \[ \sin(\pi - \theta) = \sin\theta \] for every angle 胃.

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