Bài 40 trang 213 SGK Đại số 10 Nâng cao
Chứng minh rằng: ...
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 }},,)
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