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a) Áp dụng tính chất của tỉ số lượng giác ta có:
+) Sin2α + Cos2α=1
hay \(\left(\dfrac{1}{3}\right)^2\)+Cos2α=1
\(\dfrac{1}{9}\)+Cos2α=1
Cos2α=\(\dfrac{8}{9}\)
⇒Cos α=\(\sqrt{\dfrac{8}{9}}\)=\(\dfrac{2\sqrt{2}}{3}\)
+) \(\tan\alpha=\dfrac{\sin\alpha}{\cos\alpha}=\dfrac{\dfrac{1}{3}}{\dfrac{2\sqrt{2}}{3}}=\dfrac{\sqrt{2}}{4}\)
+)\(\cot\alpha=\dfrac{\cos\alpha}{\sin\alpha}=\dfrac{\dfrac{2\sqrt{2}}{3}}{\dfrac{1}{3}}\)=\(2\sqrt{2}\)
1. \(\frac{cos\alpha+sin\alpha}{cos\alpha-sin\alpha}=\frac{1+\frac{sin\alpha}{cos\alpha}}{1-\frac{sin\alpha}{cos\alpha}}=\frac{1+\frac{1}{2}}{1-\frac{1}{2}}=3\)
2. \(cos\beta=2sin\beta\Rightarrow cos^2\beta=4sin^2\beta\). Do \(cos^2\beta+sin^2\beta=1\Rightarrow5sin^2\beta=1\Rightarrow sin\beta=\frac{1}{\sqrt{5}}\)
\(\Rightarrow cos\beta=\frac{2}{\sqrt{5}}\). Vậy \(sin\beta.cos\beta=\frac{2}{5}\)
3. a. Nhân chéo ra được hệ thức \(sin^2\alpha+cos^2\alpha=1\)
b. Chú ý \(cot^2\alpha=\frac{cos^2\alpha}{sin^2\alpha}\)
a) tan a < tan b
b) cot a > cot b
\(a.tan\alpha=\dfrac{sin\alpha}{cos\alpha}< tan\beta=\dfrac{sin\beta}{cos\beta}\)
\(b.cot\alpha=\dfrac{cos\alpha}{sin\alpha}>cot\beta=\dfrac{cos\beta}{sin\beta}\)
a)<
b)>
a) tan a < tan b
b) cot a > cot b
a) tan α < tan β
b) cot α > cot β
a, tan a < tan b
b, cot a > cot b
a) tan α= tan β
b) cot α> cot β
a) tan a < tan b
b) cot a > cot b
a, tan a = sin a/cos a< sin b/cos b = tan b
b cot a > cot b
a) tan a = tanβ
b) cot a > cot β
a, tan a < tan b
b, cot a > cot b
a)tan a < tan β
b)cot a > cot β
a, tan a< tan b
b, cot a < cot b
a) \tan\alpha=\dfrac{\sin\alpha}{\cos\alpha}< \dfrac{\sin\beta}{\cos\beta}=\tan\beta.tanα=cosαsinα<cosβsinβ=tanβ.
b) \cot \alpha > \cot \betacotα>cotβ.
a,tan α =\(\dfrac{sin^{ }a}{cos^{ }a}< \dfrac{sin^{ }\beta}{cos^{ }\beta}=tan^{ }\beta\)
b,\(cot^{ }a>cot^{ }\beta\)
\(a) \tan\alpha=\dfrac{\sin\alpha}{\cos\alpha}< \dfrac{\sin\beta}{\cos\beta}=\tan\beta.\)
\(b) \cot \alpha > \>cotβ.\)
a) tan α < tan β
b)cot α > cot β.
a:tan a < tan b
b: cos a < cos b
a) tanα<tanβ
b) cotα>cotβ
a ) tan α = \(\dfrac{\sin\alpha}{\cos\alpha}\) ≤ \(\dfrac{\sin\beta}{\cos\beta}=\tan\beta\)
b ) cot α ≥ cot β