In trigonometry, the angle addition and subtraction formulas provide a way to express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. These formulas are incredibly useful for simplifying expressions and solving trigonometric equations.
Angle Addition and Subtraction Formulas for Sine and Cosine
The angle addition and subtraction formulas for sine and cosine are as follows:
\[
\sin(A + B) = \sin A \cos B + \cos A \sin B
\]
\[
\sin(A – B) = \sin A \cos B – \cos A \sin B
\]
\[
\cos(A + B) = \cos A \cos B – \sin A \sin B
\]
\[
\cos(A – B) = \cos A \cos B + \sin A \sin B
\]
Using the Formulas to Solve for Specific Angles
Solving for \( \tan\left(\frac{\pi}{12}\right) \)
Step 1: Find two angles on the unit circle that you can add or subtract to get \(\frac{\pi}{12}\)
\[
\tan(\frac{\pi}{12}) = \tan(\frac{4\pi}{12} – \frac{3\pi}{12}) = \tan(\frac{\pi}{3} – \frac{\pi}{4})
\]
Step 2: Apply the fact \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).
\[
\tan(\frac{\pi}{3} – \frac{\pi}{4}) = \frac{\sin(\frac{\pi}{3} – \frac{\pi}{4})}{\cos(\frac{\pi}{3} – \frac{\pi}{4})}
\]
Step 3: Apply the angle subtraction formulas for sine and cosine:
\[
\sin(\frac{\pi}{3} – \frac{\pi}{4}) = \sin\left(\frac{\pi}{3}\right) \cos\left(\frac{\pi}{4}\right) – \cos\left(\frac{\pi}{3}\right) \sin\left(\frac{\pi}{4}\right)
\]
\[
\cos(\frac{\pi}{3} – \frac{\pi}{4}) = \cos\left(\frac{\pi}{3}\right) \cos\left(\frac{\pi}{4}\right) + \sin\left(\frac{\pi}{3}\right) \sin\left(\frac{\pi}{4}\right)
\]
Step 4: Use the unit circle to replace the sines and cosines with their numerical values:
\[
\sin(\frac{\pi}{3} – \frac{\pi}{4}) = \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2} – \frac{1}{2} \times \frac{\sqrt{2}}{2}
\]
\[
\cos(\frac{\pi}{3} – \frac{\pi}{4}) = \frac{1}{2} \times \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2}
\]
Step 5: Simplify to find \( \tan\left(\frac{\pi}{12}\right) \):
\[
\tan\left(\frac{\pi}{12}\right) = \frac{\sqrt{6} – \sqrt{2}}{4}
\]
Conclusion
The angle addition and subtraction formulas are powerful tools in trigonometry. They not only simplify complex trigonometric expressions but also pave the way for solving intricate problems, much like the ones we’ve tackled today. As St. Augustine once said, “The world is a book, and those who do not travel read only one page.” Similarly, the realm of trigonometry is vast, and these formulas are but stepping stones in understanding this fascinating subject.
