1. **Problem Statement:** Simplify the trigonometric expressions without using a calculator. 2. **Formula and Rules:** Use angle sum/difference identities and reference angles in the unit circle. 3. **Example 1: Simplify $\cos\left(\frac{\pi}{12}\right)$** - Note that $\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$. - Use the cosine difference identity: $$\cos(a - b) = \cos a \cos b + \sin a \sin b$$ - Substitute: $$\cos\left(\frac{\pi}{12}\right) = \cos\left(\frac{\pi}{3}\right)\cos\left(\frac{\pi}{4}\right) + \sin\left(\frac{\pi}{3}\right)\sin\left(\frac{\pi}{4}\right)$$ - Use known values: $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$, $\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$, $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$, $\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$. - Calculate: $$\frac{1}{2} \times \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{2} + \sqrt{6}}{4}$$ 4. **Example 2: Simplify $\sin\left(\frac{5\pi}{3}\right)$** - Recognize $\frac{5\pi}{3} = 2\pi - \frac{\pi}{3}$. - Use the identity: $$\sin(2\pi - x) = -\sin x$$ - So, $$\sin\left(\frac{5\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}$$ 5. **Example 3: Simplify $\cot\left(\frac{11\pi}{12}\right)$** - Note $\frac{11\pi}{12} = \pi - \frac{\pi}{12}$. - Use the identity: $$\cot(\pi - x) = -\cot x$$ - So, $$\cot\left(\frac{11\pi}{12}\right) = -\cot\left(\frac{\pi}{12}\right)$$ - Calculate $\cot\left(\frac{\pi}{12}\right) = \frac{\cos\left(\frac{\pi}{12}\right)}{\sin\left(\frac{\pi}{12}\right)}$. - From example 1, $\cos\left(\frac{\pi}{12}\right) = \frac{\sqrt{2} + \sqrt{6}}{4}$. - Use sine difference identity for $\sin\left(\frac{\pi}{12}\right) = \sin\left(\frac{\pi}{3} - \frac{\pi}{4}\right)$: $$\sin(a - b) = \sin a \cos b - \cos a \sin b$$ $$= \sin\left(\frac{\pi}{3}\right)\cos\left(\frac{\pi}{4}\right) - \cos\left(\frac{\pi}{3}\right)\sin\left(\frac{\pi}{4}\right)$$ $$= \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2} - \frac{1}{2} \times \frac{\sqrt{2}}{2} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}$$ - Therefore, $$\cot\left(\frac{\pi}{12}\right) = \frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{\frac{\sqrt{6} - \sqrt{2}}{4}} = \frac{\sqrt{2} + \sqrt{6}}{\sqrt{6} - \sqrt{2}}$$ - Rationalize denominator: $$\frac{\sqrt{2} + \sqrt{6}}{\sqrt{6} - \sqrt{2}} \times \frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}} = \frac{(\sqrt{2} + \sqrt{6})(\sqrt{6} + \sqrt{2})}{(\sqrt{6})^2 - (\sqrt{2})^2} = \frac{(\sqrt{2} + \sqrt{6})(\sqrt{6} + \sqrt{2})}{6 - 2} = \frac{(\sqrt{2} + \sqrt{6})(\sqrt{6} + \sqrt{2})}{4}$$ - Expand numerator: $$\sqrt{2} \times \sqrt{6} + \sqrt{2} \times \sqrt{2} + \sqrt{6} \times \sqrt{6} + \sqrt{6} \times \sqrt{2} = \sqrt{12} + 2 + 6 + \sqrt{12} = 2\sqrt{3} + 2 + 6 + 2\sqrt{3} = 8 + 4\sqrt{3}$$ - So, $$\cot\left(\frac{\pi}{12}\right) = \frac{8 + 4\sqrt{3}}{4} = 2 + \sqrt{3}$$ - Finally, $$\cot\left(\frac{11\pi}{12}\right) = -\cot\left(\frac{\pi}{12}\right) = -(2 + \sqrt{3})$$ **Summary:** - Use angle sum/difference identities and reference angles. - Express angles as sums or differences of known angles. - Use unit circle values for sine and cosine. - Rationalize denominators when needed. This approach avoids calculators by relying on known exact values and identities.