Тригонометрични неравенства

k \in \Z

Оттук нататък:

sin(x)

https://www.desmos.com/calculator/ehqn723ehe

sin(x) > a
a \geq 1 \Rightarrow x \in \empty
a < -1 \Rightarrow \forall x
a = -1 \Rightarrow \forall x \neq -\frac{\pi}{2} + 2k\pi
\exists \alpha: sin(\alpha) = a
\alpha + 2k\pi < x < \pi - \alpha + 2k\pi
\alpha \in \left[ -\frac{\pi}{2}; \frac{\pi}{2} \right]
sin(x) < a
a > 1 \Rightarrow \forall x
a = 1 \Rightarrow \forall x \neq \frac{\pi}{2} + 2k\pi
a \leq -1 \Rightarrow x \in \empty
\exists \alpha: sin(\alpha) = a
-\pi - \alpha + 2k\pi < x < \alpha + 2k\pi
\alpha \in \left[ -\frac{\pi}{2}; \frac{\pi}{2} \right]
cos(x)

https://www.desmos.com/calculator/bnpsmqbssw

cos(x) > a
a \geq 1 \Rightarrow x \in \empty
a < -1 \Rightarrow \forall x
a = -1 \Rightarrow x \neq \pi + 2k\pi
\exists \alpha: cos(\alpha) = a
-\alpha + 2k\pi < x < \alpha + 2k\pi
\alpha \in \left[ 0; \pi \right]
cos(x) < a
a > 1 \Rightarrow \forall x
a = 1 \Rightarrow x \neq 2k\pi
a \leq -1 \Rightarrow x \in \empty
\exists \alpha: cos(\alpha) = a
\alpha + 2k\pi < x < 2\pi - \alpha + 2k\pi
\alpha \in \left[ 0; \pi \right]
tg(x)
a \in (-\infty; +\infty)
\exists \alpha: tg(\alpha) = a
\alpha \in \left( -\frac{\pi}{2}; \frac{\pi}{2} \right)

https://www.desmos.com/calculator/wybjfqls9b

tg(x) > a
tg(x) < a
\alpha + k\pi < x < \frac{\pi}{2} + k\pi
-\frac{\pi}{2} + k\pi < x < \alpha + k\pi
cotg(x)
a \in (-\infty; +\infty)
\exists \alpha: cotg(\alpha) = a
\alpha \in \left( 0; \pi \right)

https://www.desmos.com/calculator/pgoiy2rvns

cotg(x) > a
\alpha + k\pi < x < \pi + k\pi
cotg(x) < a
k\pi < x < \alpha + k\pi

Тригонометрични неравенства

By Alexander Stoichkov

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Тригонометрични неравенства

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