Cho tan x + cot x = 4. Tính sin x, cos x, tan x, cot x

Đề bài. Cho tan x + cot x = 4. Tính sin x, cos x, tan x, cot x.

Trả lời

tan x + cot x = 4

⇒ $\tan x+\frac{1}{\tan x}=4(\tan x\ne 0\iff x\ne 0)$

⇔ tan²x – 4tanx + 1 = 0

⇔ $[\begin{array}{c}\tan x=2+\sqrt{3}\\ tanx=2-\sqrt{3}\end{array}\iff [\begin{array}{c}x={75}^{\circ }\\ x = 15°\end{array}$

Nếu x = 75° thì sinx = $\frac{\sqrt{6}\mathrm{ }+\sqrt{2}}{4};\cos x=\frac{\sqrt{6}-\sqrt{2}}{4};\cot x=\frac{1}{2+\sqrt{3}}$

Nếu x = 15° thì sinx = $\frac{\sqrt{6}\mathrm{ }-\sqrt{2}}{4};\cos x=\frac{\sqrt{6}+\sqrt{2}}{4};\cot x=\frac{1}{2-\sqrt{3}}$.