Question

Use Identities to find the exact value. 24) cos(-75°)

Answer 1

Answer:

$\frac{1}{4}$( $\sqrt{6}$ - $\sqrt{2}$)

Step-by-step explanation:

Using the addition formula for cosine

cos(x - y) = cosxcosy + sinxsiny

and the exact values

sin45° = cos45° = $\frac{1}{\sqrt{2} }$

cos60° = $\frac{1}{2}$, sin60° = $\frac{\sqrt{3} }{2}$

Note that

cos(- 75)° = cos(45 - 120)°, thus

cos(45 - 120)°

= cos45° cos120° + sin45° sin120°

= cos45° ( - cos60°) + sin45° sin60°

= $\frac{1}{\sqrt{2} }$ × - $\frac{1}{2}$ + $\frac{1}{\sqrt{2} }$ × $\frac{\sqrt{3} }{2}$

= - $\frac{1}{2\sqrt{2} }$ + $\frac{\sqrt{3} }{2\sqrt{2} }$

= $\frac{\sqrt{3}-1 }{2\sqrt{2} }$ × $\frac{\sqrt{2} }{\sqrt{2} }$

= $\frac{\sqrt{2}(\sqrt{3}-1) }{4}$

= $\frac{1}{4}$( $\sqrt{6}-\sqrt{2}$)