What is the exact value of Cosine StartFraction 7 pi Over 12 EndFraction?

Question

asked Nov 9, 2021 219k views

2 Answers

Zehnpaard · Answer 1 · 2021-11-14T10:13:18+0000

Answer:

(1)/(4) (
√(2) -
√(6) )

Explanation:

Using the addition formula for cosine

cos(a + b) = cosacosb - sinasinb

and the exact values

cos
$(\pi )/(3)$ =
(1)/(2) , sin
$(\pi )/(3)$ =
(√(3) )/(2) , cos
$(\pi )/(4)$ = sin
$(\pi )/(4)$ =
(√(2) )/(2)

Note
$(7\pi )/(12)$ =
$(\pi )/(3)$ +
$(\pi )/(4)$ , thus

cos
$(7\pi )/(12)$ = cos(
$(\pi )/(3)$ +
$(\pi )/(4)$ )

= cos
$(\pi )/(3)$ cos
$(\pi )/(4)$ - sin
$(\pi )/(3)$ sin
$(\pi )/(4)$

= (
(1)/(2) ×
(√(2) )/(2) ) - (
(√(3) )/(2) ×
)

=
(√(2) )/(4) -
(√(6) )/(4)

=
(1)/(4) (
√(2) -
√(6) )