Determine the exact value of each expression. Simplify where possible. a) sin(π/3) * tan(π/6) / cos(π/4) b) tan(π/4) + tan(π/6) - tan(π/3)

Steps to Solve

1. Evaluate the trigonometric functions For part (a), we need to find the values of the following:

   • $\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$
   • $\tan(\frac{\pi}{6}) = \frac{1}{\sqrt{3}}$
   • $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$

2. Substitute the values into the expression Now, substitute these values into the expression for part (a): $$ \frac{\sin(\frac{\pi}{3}) \cdot \tan(\frac{\pi}{6})}{\cos(\frac{\pi}{4})} = \frac{\left(\frac{\sqrt{3}}{2}\right) \cdot \left(\frac{1}{\sqrt{3}}\right)}{\frac{\sqrt{2}}{2}} $$

3. Simplify the expression Simplifying gives us: $$ \frac{\frac{\sqrt{3}}{2\sqrt{3}}}{\frac{\sqrt{2}}{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} $$

4. Evaluate the trigonometric functions for part (b) For part (b), find the values of the following:

   • $\tan(\frac{\pi}{4}) = 1$
   • $\tan(\frac{\pi}{6}) = \frac{1}{\sqrt{3}}$
   • $\tan(\frac{\pi}{3}) = \sqrt{3}$

5. Substitute values into the expression for part (b) Substitute into the expression: $$ \tan(\frac{\pi}{4}) + \tan(\frac{\pi}{6}) - \tan(\frac{\pi}{3}) = 1 + \frac{1}{\sqrt{3}} - \sqrt{3} $$

6. Combine the terms To combine the terms into a single expression, we get: $$ 1 + \frac{1}{\sqrt{3}} - \sqrt{3} = 1 - \sqrt{3} + \frac{1}{\sqrt{3}} $$ To write it over a common denominator: $$ = \frac{\sqrt{3}}{\sqrt{3}} + \frac{1}{\sqrt{3}} - \frac{3}{\sqrt{3}} = \frac{\sqrt{3} + 1 - 3}{\sqrt{3}} = \frac{\sqrt{3} - 2}{\sqrt{3}} $$