Bài 51 trang 216 SGK Đại số 10 Nâng cao
Chứng minh rằng nếu ∝ + β + γ = π thì ...
Chứng minh rằng nếu ∝ + β + γ = π thì
Chứng minh rằng nếu (∝ + β + γ = π) thì
a) (sin alpha + sin eta + sin gamma = 4cos {alpha over 2}cos {eta over 2}cos {gamma over 2})
b) (cos alpha + cos eta + cos gamma = 1 + 4sin {alpha over 2}sin {eta over 2}sin {gamma over 2})
c) (sin2∝ + sin2β + sin2γ = 4sin∝ sinβ sin γ)
d) (co{s^2} propto + { m{ }}co{s^2}eta + co{s^2}gamma { m{ }}= 1 – 2cos∝ cosβ cosγ)
Đáp án
a) Ta có:
(eqalign{
& sin alpha + sin eta + sin gammacr& = sin alpha + 2sin {{eta + gamma } over 2}cos {{eta - gamma } over 2} cr
& = sin alpha + 2sin {{pi - alpha } over 2}cos {{eta - gamma } over 2} cr&= 2sin {alpha over 2}cos {alpha over 2} + 2cos {alpha over 2} cos {{eta - gamma } over 2} cr
& = 2cos {alpha over 2}(sin {alpha over 2} + cos {{eta - gamma } over 2})cr& = 2cos {alpha over 2}{
m{[sin}}{{pi - (eta + gamma )} over 2} + cos{{eta - gamma } over 2}{
m{]}} cr
& = 2cos {alpha over 2}(cos{{eta + gamma } over 2} + cos {{eta - gamma } over 2}) cr
& =4cos {alpha over 2}cos {eta over 2}cos {gamma over 2} cr} )
b) Ta có:
(eqalign{
& cos alpha + cos eta + cos gamma cr&= 2cos {{alpha + eta } over 2}cos {{alpha - eta } over 2} + 1 - 2sin {{2gamma } over 2} cr
& = 2cos ({pi over 2} - {gamma over 2})cos{{alpha - eta } over 2} + 1 - 2{sin ^2}{gamma over 2} cr&= 1 + 2sin {gamma over 2}(cos{{alpha - eta } over 2} - sin {gamma over 2}) cr
& = 1 + 2sin {gamma over 2}(cos{{alpha - eta } over 2} - cos{{alpha + eta } over 2}) cr
& = 1 + 4sin {alpha over 2}sin {eta over 2}sin {gamma over 2} cr} )
c) (sin2∝ + sin2β + sin2γ)
(= 2sin (∝ + β)cos(∝ - β ) + 2sinγcosγ)
(= 2sinγ (cos(∝ - β ) - cos(∝ + β)) )
(= 4sin∝ sinβ sin γ)
d) Ta có:
(eqalign{
& co{s^2} propto + {
m{ }}co{s^2}eta + co{s^2}gamma {
m{ }} cr
& {
m{ = }}{{1 + cos 2alpha } over 2} + {{1cos 2eta } over 2} + {cos ^2}gamma cr
& = 1 + {1 over 2}(cos2alpha + cos 2eta ) + {cos ^2}gamma cr
& = 1 + cos (alpha + eta )cos(alpha - eta ) + {cos ^2}gamma cr
& = 1 + cos gamma (cos gamma - cos (alpha - eta )) cr&= 1 - cos gamma {
m{[cos(}}alpha {
m{ + }}eta {
m{) + cos(}}alpha {
m{ - }}eta ){
m{]}} cr
& = {
m{ }}1{
m{ }}-{
m{ }}2cos propto {
m{ }}coseta {
m{ }}cosgamma cr} )
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