Question

examples: simplify and evaluate a trig expression
example: use double angle identities to
simplify and then evaluate.
\\( \cos ^ { 2 } \left( \frac { \pi } { 12 } \
ight) - \sin ^ { 2 } \left( \frac { \pi } { 12 } \
ight) = \cos \left( \frac { 2 \pi } { 12 } \
ight) = \cos \left( \frac { \pi } { 6 } \
ight) = \frac { \sqrt { 3 } } { 2 } \\)
\\( 2 \sin \left( \frac { \pi } { 4 } \
ight) \cos \left( \frac { \pi } { 4 } \
ight) = \sin \left( \frac { 2 \pi } { 4 } \
ight) = \sin \left( \frac { \pi } { 2 } \
ight) = 1 \\)
\\( 2 \cos ^ { 2 } \left( \frac { \pi } { 2 } \
ight) - 1 \\)
\\( \cos ( 2 a ) = \cos ^ { 2 } a - \sin ^ { 2 } a \\)
\\( \cos ( 2 a ) = 1 - 2 \sin ^ { 2 } a \\)
\\( \cos ( 2 a ) = 2 \cos ^ { 2 } a - 1 \\)
\\( \sin ( 2 a ) = 2 \sin a \cos a \\)
\\( \tan ( 2 a ) = \frac { 2 \tan a } { 1 - \tan ^ { 2 } a } \\)
simplify without a calculator:
\\( 2 \sin \left( 22.5 ^ { \circ } \
ight) \cos \left( 22.5 ^ { \circ } \
ight) \\)
\\( = \\)
\\( = \\)

Explanation:

Step1: Apply double - angle formula

The double - angle formula for sine is \(\sin(2A)=2\sin A\cos A\).
In the expression \(2\sin(22.5^{\circ})\cos(22.5^{\circ})\), let \(A = 22.5^{\circ}\). Then \(2\sin(22.5^{\circ})\cos(22.5^{\circ})=\sin(2\times22.5^{\circ})\).

Step2: Calculate the angle

\(2\times22.5^{\circ}=45^{\circ}\). So, \(\sin(2\times22.5^{\circ})=\sin(45^{\circ})\).

Step3: Evaluate the sine value

We know that \(\sin(45^{\circ})=\frac{\sqrt{2}}{2}\).

Answer:

\(\sin(45^{\circ})\); \(\frac{\sqrt{2}}{2}\)