Use the appropriate compound angle formula to determine the exact value of each expression. a) sin(π + π/6) b) cos(π - π/4) c) tan(π/4 + π) d) sin(-π/2 + π/3) e) tan(π/3 - π/6) f)... Use the appropriate compound angle formula to determine the exact value of each expression. a) sin(π + π/6) b) cos(π - π/4) c) tan(π/4 + π) d) sin(-π/2 + π/3) e) tan(π/3 - π/6) f) cos(π/2 + π/3) Read more

Steps to Solve

1. Identify the appropriate formula for sine To find ( \sin\left(\pi + \frac{\pi}{6}\right) ), we can use the sine addition formula: $$ \sin(a + b) = \sin a \cos b + \cos a \sin b $$

Here, ( a = \pi ) and ( b = \frac{\pi}{6} ).

Calculate:

• ( \sin(\pi) = 0 )
• ( \cos(\pi) = -1 )
• ( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} )
• ( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} )

Then substitute into the formula: $$ \sin\left(\pi + \frac{\pi}{6}\right) = 0 \cdot \frac{\sqrt{3}}{2} + (-1) \cdot \frac{1}{2} = -\frac{1}{2} $$

2. Determine cosine using subtraction formula For ( \cos\left(\pi - \frac{\pi}{4}\right) ), use the cosine subtraction formula: $$ \cos(a - b) = \cos a \cos b + \sin a \sin b $$

Here, ( a = \pi ) and ( b = \frac{\pi}{4} ).

Calculate:

• ( \cos(\pi) = -1 )
• ( \sin(\pi) = 0 )
• ( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} )
• ( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} )

Substitute: $$ \cos\left(\pi - \frac{\pi}{4}\right) = (-1) \cdot \frac{\sqrt{2}}{2} + 0 \cdot \frac{\sqrt{2}}{2} = -\frac{\sqrt{2}}{2} $$

3. Calculate tangent for addition of angles For ( \tan\left(\frac{\pi}{4} + \pi\right) ): Use the tangent angle addition formula: $$ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} $$

Here, ( a = \frac{\pi}{4} ) and ( b = \pi ).

Calculate:

• ( \tan\left(\frac{\pi}{4}\right) = 1 )
• ( \tan(\pi) = 0 )

Then, $$ \tan\left(\frac{\pi}{4} + \pi\right) = \frac{1 + 0}{1 - 1 \cdot 0} = 1 $$

4. Determine sine for angle combination For ( \sin\left(-\frac{\pi}{2} + \frac{\pi}{3}\right) ): First, convert using the sine subtraction formula: $$ \sin(a - b) = \sin a \cos b - \cos a \sin b $$

Here, ( a = -\frac{\pi}{2} ) and ( b = \frac{\pi}{3} ).

Calculate:

• ( \sin\left(-\frac{\pi}{2}\right) = -1 )
• ( \cos\left(-\frac{\pi}{2}\right) = 0 )
• ( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} )
• ( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} )

Substitute: $$ \sin\left(-\frac{\pi}{2} + \frac{\pi}{3}\right) = -1 \cdot \frac{1}{2} - 0 \cdot \frac{\sqrt{3}}{2} = -\frac{1}{2} $$

5. Find tangent for subtraction For ( \tan\left(\frac{\pi}{3} - \frac{\pi}{6}\right) ): Use the tangent subtraction formula: $$ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} $$

Here, ( a = \frac{\pi}{3} ) and ( b = \frac{\pi}{6} ).

Calculate:

• ( \tan\left(\frac{\pi}{3}\right) = \sqrt{3} )
• ( \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} )

Substitute: $$ \tan\left(\frac{\pi}{3} - \frac{\pi}{6}\right) = \frac{\sqrt{3} - \frac{1}{\sqrt{3}}}{1 + \sqrt{3} \cdot \frac{1}{\sqrt{3}}} = \frac{\sqrt{3} - \frac{1}{\sqrt{3}}}{2} = \frac{\frac{3}{\sqrt{3}} - \frac{1}{\sqrt{3}}}{2} = \frac{\frac{2}{\sqrt{3}}}{2} = \frac{1}{\sqrt{3}} $$

6. Calculate cosine for angle addition For ( \cos\left(\frac{\pi}{2} + \frac{\pi}{3}\right) ): Use the cosine addition formula: $$ \cos(a + b) = \cos a \cos b - \sin a \sin b $$

Here, ( a = \frac{\pi}{2} ) and ( b = \frac{\pi}{3} ).

Calculate:

• ( \cos\left(\frac{\pi}{2}\right) = 0 )
• ( \sin\left(\frac{\pi}{2}\right) = 1 )
• ( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} )
• ( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} )

Substitute: $$ \cos\left(\frac{\pi}{2} + \frac{\pi}{3}\right) = 0 \cdot \frac{1}{2} - 1 \cdot \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2} $$