Introduction
The addition and subtraction formulas for sine and cosine are powerful tools in trigonometry. These formulas allow us to find the sine and cosine of the sum or difference of two angles.
Formulas
- Sine Addition Formula:$$ \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) $$
- Sine Subtraction Formula:$$ \sin(A – B) = \sin(A)\cos(B) – \cos(A)\sin(B) $$
- Cosine Addition Formula:$$ \cos(A + B) = \cos(A)\cos(B) – \sin(A)\sin(B) $$
- Cosine Subtraction Formula:$$ \cos(A – B) = \cos(A)\cos(B) + \sin(A)\sin(B) $$
Worked-out Examples
Example 1: \( \sin\left(\frac{\pi}{12}\right) \)
Using the sine subtraction formula, we get:
$$ \sin\left(\frac{\pi}{12}\right) = \sin\left(\frac{\pi}{4} – \frac{\pi}{6}\right) $$
$$ = \sin\left(\frac{\pi}{4}\right)\cos\left(\frac{\pi}{6}\right) – \cos\left(\frac{\pi}{4}\right)\sin\left(\frac{\pi}{6}\right) $$
$$ = \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} – \frac{\sqrt{2}}{2} \times \frac{1}{2} $$
$$ = \frac{\sqrt{6} – \sqrt{2}}{4} $$
Example 2: \( \cos\left(\frac{\pi}{12}\right) \)
Using the cosine subtraction formula, we get:
$$ \cos\left(\frac{\pi}{12}\right) = \cos\left(\frac{\pi}{4} – \frac{\pi}{6}\right) $$
$$ = \cos\left(\frac{\pi}{4}\right)\cos\left(\frac{\pi}{6}\right) + \sin\left(\frac{\pi}{4}\right)\sin\left(\frac{\pi}{6}\right) $$
$$ = \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \times \frac{1}{2} $$
$$ = \frac{\sqrt{6} + \sqrt{2}}{4} $$
Example 3: \( \tan\left(\frac{\pi}{12}\right) \)
Using the sine and cosine formulas, we get:
$$ \tan\left(\frac{\pi}{12}\right) = \frac{\sin\left(\frac{\pi}{12}\right)}{\cos\left(\frac{\pi}{12}\right)} $$
$$ = \frac{\frac{\sqrt{6} – \sqrt{2}}{4}}{\frac{\sqrt{6} + \sqrt{2}}{4}} $$
$$ = \frac{\sqrt{6} – \sqrt{2}}{\sqrt{6} + \sqrt{2}} $$
$$ = \frac{\sqrt{6} – \sqrt{2}}{\sqrt{6} + \sqrt{2}} \times \frac{\sqrt{6} – \sqrt{2}}{\sqrt{6} – \sqrt{2}} $$
$$ = \frac{(\sqrt{6} – \sqrt{2})^2}{(\sqrt{6})^2 – (\sqrt{2})^2} $$
$$ = \frac{6 – 2\sqrt{12} + 2}{4} $$
$$ = \frac{8 – 2\sqrt{12}}{4} $$
$$ = \frac{2(4 – \sqrt{12})}{4} $$
$$ = \frac{4 – \sqrt{12}}{2} $$
Practice Problems
Use the addition and subtraction formulas to find the following:
- $$ \sin\left(\frac{\pi}{2} + \frac{\pi}{6}\right) $$
- $$ \cos\left(\frac{\pi}{3} – \frac{\pi}{4}\right) $$
