Có \(a\) thuộc góc phần tư thứ III -> sin\(a\) < 0

+) sin\(a\)=-\(\sqrt{1-cos^2a}\)=-\(\sqrt{1-\left(\dfrac{-12}{13}\right)^2}\)=\(\dfrac{-5}{13}\)

\(cos2a=cos^2a-sin^2a\)=\(\left(\dfrac{-12}{13}\right)^2-\left(\dfrac{-5}{13}\right)^2=\dfrac{119}{169}\)

b) Ta có: \(\sin^2\alpha+\cos^2\alpha=1\)

\(\Leftrightarrow\cos^2\alpha=\dfrac{16}{25}\)

hay \(\cos\alpha=\dfrac{4}{5}\)

Ta có: \(A=5\cdot\sin^2\alpha+6\cdot\cos^2\alpha\)

\(=5\cdot\left(\dfrac{3}{5}\right)^2+6\cdot\left(\dfrac{4}{5}\right)^2\)

\(=5\cdot\dfrac{9}{25}+6\cdot\dfrac{16}{25}\)

\(=\dfrac{141}{25}\)

c) Ta có: \(\tan\alpha=\dfrac{1}{\cot\alpha}=\dfrac{1}{\dfrac{4}{3}}=\dfrac{3}{4}\)

\(D=\dfrac{\sin\alpha+\cos\alpha}{\sin\alpha-\cos\alpha}\)

\(=\dfrac{\dfrac{9}{16}+\dfrac{16}{9}}{\dfrac{9}{16}-\dfrac{16}{9}}=-\dfrac{337}{175}\)

ta có : \(B=\dfrac{cot\alpha-cos\alpha}{cos^3\alpha}\) \(\left(đk:cosx\ne0\right)\)

\(\Leftrightarrow B=\dfrac{cos\alpha\left(\dfrac{1}{sin\alpha}-1\right)}{cos^3\alpha}\Leftrightarrow B=\dfrac{\dfrac{1}{sin\alpha}-1}{cos^2\alpha}\)

\(\Leftrightarrow B=\dfrac{\dfrac{1}{sin\alpha}-1}{1-sin^2\alpha}=\dfrac{\dfrac{13}{5}-1}{1-\left(\dfrac{5}{13}\right)^2}=\dfrac{169}{90}\)

vậy \(B=\dfrac{169}{90}\) khi \(sin\alpha=\dfrac{5}{13}\)

Ta có:

 \(\begin{array}{l}\sin \left( { - \frac{{15\pi }}{2} - \alpha } \right) - \cos \left( {13\pi  + \alpha } \right) =  \sin \left( { -\frac{{16\pi }}{2} +\frac{{\pi }}{2}  + \alpha } \right) - \cos \left( {12\pi  + \pi + \alpha } \right) =  \sin \left( {-8\pi  + \frac{\pi }{2} - \alpha } \right) - \cos \left( { \pi + \alpha } \right) \\ = \sin \left( {\frac{\pi }{2} - \alpha } \right) + \cos \left( \alpha  \right) = \cos \left( \alpha  \right) + \cos \left( \alpha  \right) = 2\cos \left( \alpha  \right) = 2.\left( { - \frac{5}{{13}}} \right) = \frac{{ - 10}}{{13}}\end{array}\)

\(\dfrac{sina+cosa}{sina-cosa}=\dfrac{\dfrac{sina+cosa}{cosa}}{\dfrac{sina-cosa}{cosa}}=\dfrac{tana+1}{tana-1}=\dfrac{3}{1}=3\)

Có \(\dfrac{sin\alpha}{cos\alpha}=tan\alpha=2\)\(\Rightarrow sin\alpha=2cos\alpha\)

\(\dfrac{sin\alpha+cos\alpha}{sin\alpha-cos\alpha}=\dfrac{2cos\alpha+cos\alpha}{2cos\alpha-cos\alpha}=\dfrac{3cos\alpha}{cos\alpha}=3\)

\(A=sin\left(\dfrac{\pi}{2}-\alpha+2\pi\right)+cos\left(\pi+\alpha+12\pi\right)-3sin\left(\alpha-\pi-4\pi\right)\)

\(=sin\left(\dfrac{\pi}{2}-\alpha\right)+cos\left(\pi+\alpha\right)-3sin\left(\alpha-\pi\right)\)

\(=cos\alpha-cos\alpha+3sin\left(\pi-\alpha\right)\)\(=3sin\alpha\)

\(B=sin\left(x+\dfrac{\pi}{2}+42\pi\right)+cos\left(x+\pi+2016\pi\right)+sin^2\left(x+\pi+32\pi\right)+sin^2\left(x-\dfrac{\pi}{2}-2\pi\right)+cos\left(x-\dfrac{\pi}{2}+2\pi\right)\)

\(=sin\left(x+\dfrac{\pi}{2}\right)+cos\left(x+\pi\right)+sin^2\left(x+\pi\right)+sin^2\left(x-\dfrac{\pi}{2}\right)+cos\left(x-\dfrac{\pi}{2}\right)\)

\(=cosx-cosx+sin^2x+cos^2x+sinx\)

\(=1+sinx\)

\(C=sin\left(x+\dfrac{\pi}{2}+1008\pi\right)+2sin^2\left(\pi-x\right)+cos\left(x+\pi+2018\pi\right)+cos2x+sin\left(x+\dfrac{\pi}{2}+4\pi\right)\)

\(=sin\left(x+\dfrac{\pi}{2}\right)+2sin^2\left(\pi-x\right)+cos\left(x+\pi\right)+cos2x+sin\left(x+\dfrac{\pi}{2}\right)\)

\(=cosx+2sin^2x-cosx+1-2sin^2x+cosx\)

\(=1+cosx\)

Câu 2:

\(A=2\cdot\dfrac{1}{2}+3\cdot\dfrac{1}{2}+1=1+1+1=3\)

Bài 3:

\(cos^2a=1-\left(\dfrac{12}{13}\right)^2=\dfrac{25}{169}\)

mà cosa>0

nên cosa=5/13

=>tan a=12/5; cot a=5/12

Câu 4: \(sin^2a=1-\dfrac{1}{4}=\dfrac{3}{4}\)

mà sina <0

nên sin a=-căn 3/2

=>tan a=-căn 3

\(A=-\dfrac{\sqrt{3}}{2}+\dfrac{1}{2}\cdot\left(-\sqrt{3}\right)=-\sqrt{3}\)

Lời giải:

a) \(A=5\sin ^2a+6\cos ^2a=6(\sin ^2a+\cos ^2a)-\sin ^2a\)

\(=6.1-(\frac{3}{5})^2=\frac{141}{25}\)

b)

\(\tan a=\frac{5}{12}\Leftrightarrow \frac{\sin a}{\cos a}=\frac{5}{12}\)

\(\Rightarrow \frac{\sin a}{5}=\frac{\cos a}{12}\Rightarrow \frac{\sin ^2a}{5^2}=\frac{\cos ^2a}{12^2}=\frac{\sin ^2a+\cos ^2a}{5^2+12^2}=\frac{1}{169}\)

(theo tính chất dãy tỉ số bằng nhau)

\(\Rightarrow \sin ^2a=\frac{5^2}{169}; \cos ^2a=\frac{12^2}{169}\)

Kết hợp với việc \(\sin a, \cos a\) cùng dấu (do thương của chúng dương)

\(\Rightarrow (\sin a, \cos a)=\left(\frac{5}{13}; \frac{12}{13}\right)\) hoặc \(\left(\frac{-5}{13}; \frac{-12}{13}\right)\)

Lời giải:

$\sin ^2a+\cos ^2a=1$

$\cos ^2a=1-\sin ^2a=1-(\frac{-5}{13})^2=\frac{144}{169}$

Vì $\pi < a< \frac{3\pi}{2}$ nên $\cos a< 0$

Do đó: $\cos a=-\sqrt{\frac{144}{169}}=\frac{-12}{13}$

$\sin 2a=2\sin a\cos a=2.\frac{-5}{13}.\frac{-12}{13}=\frac{120}{169}$

$\cos 2a=\cos ^2a-\sin ^2a=2\cos ^2a-1=2.\frac{144}{169}-1=\frac{119}{169}$

$\cos a=\cos ^2\frac{a}{2}-\sin ^2\frac{a}{2}$

$=1-2\sin ^2\frac{a}{2}$

$\Leftrightarrow \frac{-12}{13}=1-2\sin ^2\frac{a}{2}$

$\Rightarrow \sin ^2\frac{a}{2}=\frac{25}{26}$

Vì $\pi < a< \frac{3\pi}{2}$ nên $\sin \frac{a}{2}>0$

$\Rightarrow \sin \frac{a}{2}=\frac{5}{\sqrt{26}}$