cho cosa =3/4.Tinh A= cos(3a/2)cos(a/2)
cho \(cosa=\frac{3}{4}\). tính \(cos\frac{3a}{2}.cos\frac{a}{2}\)
\(cos\frac{3a}{2}.cos\frac{a}{2}=\frac{1}{2}\left(cos2a+cosa\right)=\frac{1}{2}\left(2cos^2a-1+cosa\right)\)
\(=\frac{1}{2}\left(2.\frac{9}{16}-1+\frac{3}{4}\right)=...\)
Đề: Rút gọn biểu thức sau: D= (1+ cosa + cos 2a + cos 3a)/ (cosa + 2 cos ²a - 1)
Lời giải:
$D=\frac{1+\cos a+2\cos ^2a-1+4\cos ^3a-3\cos a}{\cos a+2\cos ^2a-1}$
$=\frac{4\cos ^3a+2\cos ^2a-2\cos a}{\cos a+2\cos ^2a-1}$
$=\frac{2\cos a(\cos a+2\cos ^2a-1)}{\cos a+2\cos ^2a-1}$
$=2\cos a$
Giúp mình với các bạn ơi!!!!!!!!!!!!!!
Cho sina*cosa=0.22. Tính giá trị của biểu thức M=\(\sin^3a+\cos^3a-2.\sin a.\cos a\)
cho sina+cosa=5/4
a, A=sina.cosa b, B= sina-cosa c,C=sin^3a-cos^3a
help me
\(sina+cosa=\frac{5}{4}\Rightarrow\left(sina+cosa\right)^2=\frac{25}{16}\)
\(\Rightarrow sin^2a+cos^2a+2sina.cosa=\frac{25}{16}\)
\(sina.cosa=\frac{\frac{25}{16}-1}{2}=\frac{9}{32}\)
b/ \(\left(sina-cosa\right)^2=sin^2a+cos^2a-2sinacosa\)
\(\left(sina-cosa\right)^2=1-2.\frac{9}{32}=\frac{7}{16}\)
\(\Rightarrow sina-cosa=\pm\frac{\sqrt{7}}{4}\)
c/ \(sin^3a-cos^3a=\left(sina-cosa\right)\left(sin^2a+cos^2a+sina.cosa\right)\)
\(=\left(sina-cosa\right)\left(1+\frac{9}{32}\right)=\pm\frac{41\sqrt{7}}{128}\)
Dựa vào các công thức cộng đã học:
sin(a + b) = sina cosb + sinb cosa;
sin(a – b) = sina cosb - sinb cosa;
cos(a + b) = cosa cosb – sina sinb;
cos(a – b) = cosa cosb + sina sinb;
và kết quả cos π/4 = sinπ/4 = √2/2, hãy chứng minh rằng:
a) sinx + cosx = √2 cos(x - π/4);
b) sin x – cosx = √2 sin(x - π/4).
a) √2 cos(x - π/4)
= √2.(cosx.cos π/4 + sinx.sin π/4)
= √2.(√2/2.cosx + √2/2.sinx)
= √2.√2/2.cosx + √2.√2/2.sinx
= cosx + sinx (đpcm)
b) √2.sin(x - π/4)
= √2.(sinx.cos π/4 - sin π/4.cosx )
= √2.(√2/2.sinx - √2/2.cosx )
= √2.√2/2.sinx - √2.√2/2.cosx
= sinx – cosx (đpcm).
Cho sina + cosa =2. Tính sin^3a + cos^3a
ta có : \(sin^3a+cos^3a=\left(sina+cosa\right)^3-3sina.cosa\left(sina+cosa\right)\)
\(=2^3-3sina.cosa\left(2\right)=8-6sina.cosa\)
\(=11-3sin^2a-6sina.cosa-3cos^2a=11-3\left(sin+cos\right)^2=11-3.2^2=11-12=-1\)
chứng minh các đẳng thức sau :
a)\(\frac{cos\left(a-b\right)}{cos\left(a+b\right)}=\frac{cota.cotb+1}{cota.cotb-1}\)
b)\(2\left(sin^6a+cos^6a\right)+1=3\left(sin^4a+cos^4a\right)\)
c)\(\frac{tana-tanb}{cotb-cota}=tanatanb\)
d)\(\left(cotx+tanx\right)^2-\left(cotx-tanx\right)^2=4\)
e)\(\frac{sin^3a+cos^3a}{sina+cosa}=1-sinacosa\)
Lời giải:
a)
\(\frac{\cos (a-b)}{\cos (a+b)}=\frac{\cos a\cos b+\sin a\sin b}{\cos a\cos b-\sin a\sin b}=\frac{\frac{\cos a\cos b}{\sin a\sin b}+1}{\frac{\cos a\cos b}{\sin a\sin b}-1}=\frac{\cot a\cot b+1}{\cot a\cot b-1}\)
b)
\(2(\sin ^6a+\cos ^6a)+1=2(\sin ^2a+\cos ^2a)(\sin ^4a-\sin ^2a\cos ^2a+\cos ^4a)+1\)
\(=2(\sin ^4a-\sin ^2a\cos ^2a+\cos ^4a)+1\)
\(=3(\sin ^4a+\cos ^4a)-(\sin ^4a+\cos ^4a+2\sin ^2a\cos ^2a)+1\)
\(=3(\sin ^4a+\cos ^4a)-(\sin ^2a+\cos ^2a)^2+1\)
\(=3(\sin ^4a+\cos ^4a)-1^2+1=3(\sin ^4a+\cos ^4a)\)
c)
\(\frac{\tan a-\tan b}{cot b-\cot a}=\frac{\tan a-\tan b}{\frac{1}{\tan b}-\frac{1}{\tan a}}\) (nhớ rằng \(\tan x.\cot x=1\rightarrow \cot x=\frac{1}{\tan x}\) )
\(=\frac{\tan a-\tan b}{\frac{\tan a-\tan b}{\tan a\tan b}}=\tan a\tan b\)
d)
\((\cot x+\tan x)^2-(\cot x-\tan x)^2=(\cot ^2x+\tan ^2x+2\cot x\tan x)-(\cot ^2x-2\cot x\tan x+\tan ^2x)\)
\(=4\cot x\tan x=4.1=4\)
e)
\(\frac{\sin ^3a+\cos ^3a}{\sin a+\cos a}=\frac{(\sin a+\cos a)(\sin ^2a-\sin a\cos a+\cos ^2a)}{\sin a+\cos a}\)
\(=\sin ^2a-\sin a\cos a+\cos ^2a=(\sin ^2a+\cos ^2a)-\sin a\cos a=1-\sin a\cos a\)
Vậy ta có đpcm.
Chứng minh rằng:
Cos^4(a)+sin^4(a)+tan^3(a)=
Sina+cosa/cos^2(a)
rút gọn A=\(\frac{sin^3a-cos^3a}{sina-cosa}+sina+cosa\)
\(A=\frac{\left(sina-cosa\right)\left(sin^2a+cos^2a+sina.cosa\right)}{sina-cosa}+sina+cosa\)
\(=1+sina.cosa+sina+cosa\)
\(=\left(sina+1\right)\left(cosa+1\right)\)