Cho \(cot\alpha=\dfrac{a^2-b^2}{2ab}\). Trong đó \(\alpha\) là góc nhọn, a>b>0. Tính \(cos\alpha\)
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tuỳ ý giá trị biểu thức
LP
1
5 tháng 9 2022
\(1+cot^2a=\dfrac{1}{sin^2a}\)
\(\Leftrightarrow\dfrac{1}{sin^2a}=1+\dfrac{\left(a^2-b^2\right)^2}{4a^2b^2}=\dfrac{4a^2b^2+a^4-2a^2b^2+b^4}{4a^2b^2}\)
\(\Leftrightarrow sin^2a=\dfrac{4a^2b^2}{a^4+2a^2b^2+b^4}=\left(\dfrac{2ab}{\left(a^2+b^2\right)}\right)^2\)
=>\(cos^2a=\dfrac{a^4+2a^2b^2+b^4-4a^2b^2}{\left(a^2+b^2\right)^2}\)
\(\Leftrightarrow cos^2a=\dfrac{\left(a^2-b^2\right)^2}{\left(a^2+b^2\right)^2}\)
hay \(cosa=\dfrac{\left(a^2-b^2\right)}{a^2+b^2}\)
AH
Akai Haruma
Giáo viên
26 tháng 7 2021
Lời giải:
a.
$\tan a+\cot a=2\Leftrightarrow \tan a+\frac{1}{\tan a}=2$
$\Leftrightarrow \frac{\tan ^2a+1}{\tan a}=2$
$\Leftrightarrow \tan ^2a-2\tan a+1=0$
$\Leftrightarrow (\tan a-1)^2=0\Rightarrow \tan a=1$
$\cot a=\frac{1}{\tan a}=1$
$1=\tan a=\frac{\cos a}{\sin a}\Rightarrow \cos a=\sin a$
Mà $\cos ^2a+\sin ^2a=1$
$\Rightarrow \cos a=\sin a=\pm \frac{1}{\sqrt{2}}$
b.
Vì $\sin a=\cos a=\pm \frac{1}{\sqrt{2}}$
$\Rightarrow \sin a\cos a=\frac{1}{2}$
$E=\frac{\sin a.\cos a}{\tan ^2a+\cot ^2a}=\frac{\frac{1}{2}}{1+1}=\frac{1}{4}$
NV
Nguyễn Việt Lâm
Giáo viên
8 tháng 2 2022
\(A=\dfrac{\dfrac{3sina}{sina}-\dfrac{cosa}{sina}}{\dfrac{2sina}{sina}+\dfrac{cosa}{sina}}=\dfrac{3-cota}{2+cota}=\dfrac{3-3}{2+3}=0\)
\(B=\dfrac{\dfrac{sin^2a}{sin^2a}-\dfrac{3sina.cosa}{sin^2a}+\dfrac{2}{sin^2a}}{\dfrac{2sin^2a}{sin^2a}+\dfrac{sina.cosa}{sin^2a}+\dfrac{cos^2a}{sin^2a}}=\dfrac{1-3cota+2\left(1+cot^2a\right)}{2+cota+cot^2a}=\dfrac{1-3.3+2\left(1+3^2\right)}{2+3+3^2}=...\)
AM
8 tháng 2 2022
a. \(A=\dfrac{3sin\alpha-cos\alpha}{2sin\alpha+cos\alpha}=\dfrac{3\dfrac{sin\alpha}{cos\alpha}-1}{2\dfrac{sin\alpha}{cos\alpha}+1}=\dfrac{3.\dfrac{1}{3}-1}{2.\dfrac{1}{3}+1}=0\)
b.\(B=\dfrac{sin^2\alpha-3sin\alpha.cos\alpha+2}{2sin^2\alpha+sin\alpha.cos\alpha+cos^2\alpha}\)\(=\dfrac{1-\dfrac{3cos\alpha}{sin\alpha}+\dfrac{2}{sin^2\alpha}}{2+\dfrac{cos\alpha}{sin\alpha}+\dfrac{cos^2\alpha}{sin^2\alpha}}=\dfrac{1-3.3+\dfrac{2}{sin^2\alpha}}{2+3+3^2}\)
Mà \(\dfrac{cos\alpha}{sin\alpha}=3,cos^2\alpha+sin^2\alpha=1\Rightarrow sin^2\alpha=\dfrac{1}{10}\)
\(B=\dfrac{1-3.3+\dfrac{2}{\dfrac{1}{10}}}{2+3+3^2}=\dfrac{6}{7}\)
27 tháng 8 2021
a/ \(A=\frac{cot^2a-cos^2a}{cot^2a}-\frac{sina.cosa}{cota}\)
\(=\frac{\frac{cos^2a}{sin^2a}-cos^2a}{\frac{cos^2a}{sin^2a}}-\frac{sina.cosa}{\frac{cosa}{sina}}\)
\(=\left(1-sin^2a\right)-sin^2a=1\)
27 tháng 8 2021
b/ \(B=\left(cosa-sina\right)^2+\left(cosa+sina\right)^2+cos^4a-sin^4a-2cos^2a\)
\(=cos^2a-2cosa.sina+sin^2a+cos^2a+2cosa.sina+sin^2a+\left(cos^2a+sin^2a\right)\left(cos^2a-sin^2a\right)-2cos^2a\)
\(=2+\left(cos^2a-sin^2a\right)-2cos^2a\)
\(=2-sin^2a-cos^2a=2-1=1\)
BT
10 tháng 5 2017
Do \(\dfrac{\pi}{2}< \alpha< \pi\) nên \(tan\alpha< 0,cot\alpha< 0;cos\alpha< 0\).
Vì vậy: \(cos\alpha=-\sqrt{1-sin^2\alpha}=-\dfrac{\sqrt{7}}{4}\).
\(tan\alpha=\dfrac{sin\alpha}{cos\alpha}=\dfrac{3}{4}:\dfrac{-\sqrt{7}}{4}=\dfrac{-3}{\sqrt{7}}\).
\(cot\alpha=\dfrac{1}{tan\alpha}=\dfrac{-\sqrt{7}}{3}\).
\(A=\dfrac{2tan\alpha-3cot\alpha}{cos\alpha+tan\alpha}\)\(=\dfrac{2.\dfrac{-3}{\sqrt{7}}-3.\dfrac{-\sqrt{7}}{3}}{\dfrac{-\sqrt{7}}{4}+\dfrac{-3}{\sqrt{7}}}\)
\(=\dfrac{\dfrac{-6}{\sqrt{7}}+\sqrt{7}}{\dfrac{-7-12}{4\sqrt{7}}}\)\(=\dfrac{\dfrac{-6+7}{\sqrt{7}}.4\sqrt{7}}{-19}\)\(=\dfrac{\dfrac{1}{\sqrt{7}}.4\sqrt{7}}{-19}=-\dfrac{4}{19}\).
BT
10 tháng 5 2017
b) \(\dfrac{cos^2\alpha+cot^2\alpha}{tan\alpha-cot\alpha}=\dfrac{\left(-\dfrac{\sqrt{7}}{4}\right)^2+\left(\dfrac{-\sqrt{7}}{3}\right)^2}{\dfrac{-3}{\sqrt{7}}+\dfrac{\sqrt{7}}{3}}\)
\(=\dfrac{\dfrac{7}{16}+\dfrac{7}{9}}{\dfrac{-9+7}{3\sqrt{7}}}=\dfrac{\dfrac{175}{144}}{\dfrac{-2}{3\sqrt{7}}}=\dfrac{-175}{96\sqrt{7}}\).
ND
Nguyễn Đức Trí
VIP
12 tháng 9 2023
1) \(cot\alpha=\sqrt[]{5}\Rightarrow tan\alpha=\dfrac{1}{\sqrt[]{5}}\)
\(C=sin^2\alpha-sin\alpha.cos\alpha+cos^2\alpha\)
\(\Leftrightarrow C=\dfrac{1}{cos^2\alpha}\left(tan^2\alpha-tan\alpha+1\right)\)
\(\Leftrightarrow C=\left(1+tan^2\alpha\right)\left(tan^2\alpha-tan\alpha+1\right)\)
\(\Leftrightarrow C=\left(1+\dfrac{1}{5}\right)\left(\dfrac{1}{5}-\dfrac{1}{\sqrt[]{5}}+1\right)\)
\(\Leftrightarrow C=\dfrac{6}{5}\left(\dfrac{6}{5}-\dfrac{\sqrt[]{5}}{5}\right)=\dfrac{6}{25}\left(6-\sqrt[]{5}\right)\)
12 tháng 9 2023
1: \(cota=\sqrt{5}\)
=>\(cosa=\sqrt{5}\cdot sina\)
\(1+cot^2a=\dfrac{1}{sin^2a}\)
=>\(\dfrac{1}{sin^2a}=1+5=6\)
=>\(sin^2a=\dfrac{1}{6}\)
\(C=sin^2a-sina\cdot\sqrt{5}\cdot sina+\left(\sqrt{5}\cdot sina\right)^2\)
\(=sin^2a\left(1-\sqrt{5}+5\right)=\dfrac{1}{6}\cdot\left(6-\sqrt{5}\right)\)
2: tan a=3
=>sin a=3*cosa
\(1+tan^2a=\dfrac{1}{cos^2a}\)
=>\(\dfrac{1}{cos^2a}=1+9=10\)
=>\(cos^2a=\dfrac{1}{10}\)
\(B=\dfrac{3\cdot cosa-cosa}{27\cdot cos^3a+3\cdot cos^3a+2\cdot3\cdot cosa}\)
\(=\dfrac{2\cdot cosa}{30cos^3a+6cosa}=\dfrac{2}{30cos^2a+6}\)
\(=\dfrac{2}{3+6}=\dfrac{2}{9}\)
\(cot^2a=\left(\dfrac{a^2-b^2}{2ab}\right)^2\Leftrightarrow\dfrac{cos^2a}{sin^2a}=\dfrac{a^4+b^4-2a^2b^2}{4a^2b^2}\)
\(\Leftrightarrow\dfrac{cos^2a}{sin^2a}+1=\dfrac{a^4+b^4-2a^2b^2}{4a^2b^2}+1\)
\(\Leftrightarrow\dfrac{1}{sin^2a}=\dfrac{a^4+b^4+2a^2b^2}{4a^2b^2}\)
\(\Leftrightarrow sin^2a=\dfrac{4a^2b^2}{a^4+b^4+2a^2b^2}\)
\(\Leftrightarrow cos^2a=1-sin^2a=1-\dfrac{4a^2b^2}{a^4+b^4+2a^2b^2}=\dfrac{a^4+b^4-2a^2b^2}{a^4+b^4+2a^2b^2}\)
\(\Leftrightarrow cos^2a=\left(\dfrac{a^2-b^2}{a^2+b^2}\right)^2\)
\(\Leftrightarrow cosa=\dfrac{a^2-b^2}{a^2+b^2}\)
Nhìn sự khác nhau giữa dòng 2 và dòng 3 và tự suy luận đi em, rất đơn giản đúng ko?