Bài 40 trang 213 Đại số 10 Nâng cao: Chứng minh rằng:...

Chứng minh rằng:

a) (sin alpha  + cos alpha  = sqrt 2 sin (alpha  + {pi  over 4}))

b) (sin alpha  – cos alpha  = sqrt 2 sin (alpha  – {pi  over 4}))

c) ( an ({pi  over 4} – alpha ) = {{1 – an alpha } over {1 + an alpha }},,(alpha   e {pi  over 2} + kpi ;,,alpha   e {{3pi } over 4} + kpi ))

d) ( an ({pi  over 4} + alpha ) = {{1 + an alpha } over {1 – an alpha }},,(alpha   e {pi  over 2} + kpi ;,,alpha   e {pi  over 4} + kpi ))

Đáp án

a) Ta có:

(eqalign{
& sqrt 2 sin (alpha + {pi over 4}) = sqrt 2 (sin alpha cos {pi over 4} + sin {pi over 4}cos alpha ) cr
& = sqrt 2 (sin alpha {{sqrt 2 } over 2} + {{sqrt 2 } over 2}cos alpha ) cr
& = sin alpha + cos alpha cr} ) 

b) Ta có:

(eqalign{
& sqrt 2 sin (alpha – {pi over 4}) = sqrt 2 (sin alpha cos {pi over 4} – sin {pi over 4}cos alpha ) cr
& = sinalpha – cos alpha cr} ) 

c) Ta có:

( an ({pi  over 4} – alpha ) = {{ an {pi  over 4} – an alpha } over {1 + an {pi  over 4} an alpha }} = {{1 – an alpha } over {1 + an alpha }},)

d) Ta có:

( an ({pi  over 4} + alpha ) = {{ an {pi  over 4} + an alpha } over {1 – an {pi  over 4} an alpha }} = {{1 + an alpha } over {1 – an alpha }},,)