Use the addition formulas: $$\begin{array} { l }
\sin ( x + y ) = \sin x \cdot \cos y + \cos x \cdot \sin y \
\sin ( x - y ) = \sin x \cdot \cos y - \cos x \cdot \sin y \
\cos ( x + y ) = \cos x \cdot \cos y - \sin x \cdot \sin y \
\cos ( x - y ) = \cos x \cdot \cos y + \sin x \cdot \sin y
\end{array}$$
To calculate $\cos \left( \frac { \pi } { 3 } \right)$ , given that $\sin \left( \frac { \pi } { 6 } \right) = \frac { 1 } { 2 }$ and $\cos \left( \frac { \pi } { 6 } \right) = \frac { \sqrt { 3 } } { 2 }$ .

A) $\cos \left( \frac { \pi } { 3 } \right) = 0$
B) $\cos \left( \frac { \pi } { 3 } \right) = \frac { \sqrt { 3 } } { 2 }$
C) $\cos \left( \frac { \pi } { 3 } \right) = \frac { 1 } { 2 }$
D) $\cos \left( \frac { \pi } { 3 } \right) = - \frac { \sqrt { 3 } } { 2 }$
E) $\cos \left( \frac { \pi } { 3 } \right) = - \frac { 1 } { 2 }$

Question

Use the addition formulas:  $$\begin{array} { l } 
\sin ( x + y ) = \sin x \cdot \cos y + \cos x \cdot \sin y \
\sin ( x - y ) = \sin x \cdot \cos y - \cos x \cdot \sin y \
\cos ( x + y ) = \cos x \cdot \cos y - \sin x \cdot \sin y \
\cos ( x - y ) = \cos x \cdot \cos y + \sin x \cdot \sin y
\end{array}$$ 
To calculate $\cos \left( \frac { \pi } { 3 } \right)$ , given that $\sin \left( \frac { \pi } { 6 } \right) = \frac { 1 } { 2 }$ and $\cos \left( \frac { \pi } { 6 } \right) = \frac { \sqrt { 3 } } { 2 }$ .
 
A) $\cos \left( \frac { \pi } { 3 } \right) = 0$
B) $\cos \left( \frac { \pi } { 3 } \right) = \frac { \sqrt { 3 } } { 2 }$
C) $\cos \left( \frac { \pi } { 3 } \right) = \frac { 1 } { 2 }$
D) $\cos \left( \frac { \pi } { 3 } \right) = - \frac { \sqrt { 3 } } { 2 }$
E) $\cos \left( \frac { \pi } { 3 } \right) = - \frac { 1 } { 2 }$

Quizplus · Accepted Answer

Using the formula for $\cos(x + y) = \cos x \cdot \cos y - \sin x \cdot \sin y$, and setting $x = y = \frac{\pi}{6}$, we get $\cos\left(\frac{\pi}{3}\right) = \cos^2\left(\frac{\pi}{6}\right) - \sin^2\left(\frac{\pi}{6}\right) = \left(\frac{\sqrt{3}}{2}\right)^2 - \left(\frac{1}{2}\right)^2 = \frac{3}{4} - \frac{1}{4} = \frac{1}{2}$.

Use the Addition Formulas To Calculate cos⁡(π3)\cos \left( \frac { \pi } { 3 } \right)cos(3π​)

Use the Addition Formulas To Calculate $\cos \left( \frac { \pi } { 3 } \right)$