Wzory trygonometryczne
1. Jedynka tryonometryczna:
[tex]\huge\boxed{sin^2\alpha+cos^2\alpha=1}[/tex]
2. Wzory na tangens i contangens:
[tex]\huge\boxed{\begin{array}{c}tg\alpha*ctg\alpha=1\\\\tg\alpha=\dfrac{sin\alpha}{cos\alpha}\\\\ctg\alpha=\dfrac{cos\alpha}{sin\alpha}\end{array}}[/tex]
Funkcje trygonometryczne w ćwiartkach układu współrzędnych
[tex]\begin{array}{|c|c|c|c|c|}\cline{1-5}\:&I&II&III&IV\\\cline{1-5}sin\alpha&+&+&-&-\\\cline{1-5}cos\alpha&+&-&-&+\\\cline{1-5}tg\alpha&+&-&+&-\\\cline{1-5}ctg\alpha&+&-&+&-\\\cline{1-5}\end{array}[/tex]
Rozwiązanie:
[tex]\alpha \in \langle90^\circ; 180^\circ\rangle[/tex]
Kąt znajduje się w drugiej ćwiartce układu współrzędnych.
a)
[tex]sin\alpha=\dfrac{\sqrt6}4; cos\alpha, tg\alpha, ctg\alpha < 0\\\\\\\left(\dfrac{\sqrt6}4\right)^2+cos^2\alpha=1\\\\\dfrac{6}{16}+cos^2\alpha=1 /-\dfrac6{16}\\\\cos^2\alpha=\dfrac{10}{16}\\\\\boxed{cos\alpha=-\dfrac{\sqrt{10}}4}\\\\tg\alpha=\dfrac{\sqrt6}4*\left(-\dfrac{4}{\sqrt{10}}\right)\\\\tg\alpha=-\dfrac{\sqrt6}{\sqrt{10}}=-\sqrt{\dfrac{6}{10}}=-\sqrt{\dfrac35}\\\\\\tg\alpha=-\dfrac{\sqrt3}{\sqrt5}*\dfrac{\sqrt5}{\sqrt5}\\\\\boxed{tg\alpha=-\dfrac{\sqrt{15}}5}\\\\[/tex]
[tex]ctg\alpha=-\dfrac{5}{\sqrt{15}}*\dfrac{\sqrt{15}}{\sqrt{15}}=\\\\ctg\alpha=-\dfrac{5\!\!\!\!\diagup^1\sqrt{15}}{15\!\!\!\!\!\diagup_3}\\\\\boxed{ctg\alpha=-\dfrac{\sqrt{15}}3}[/tex]
b)
[tex]cos\alpha=-\frac15; sin\alpha > 0; tg\alpha, ctg\alpha < 0[/tex]
[tex]sin^2\alpha+\left(-\dfrac15\right)^2=1\\\\sin^2\alpha+\dfrac1{25}=1 /-\dfrac1{25}\\\\sin^2\alpha=\dfrac{24}{25}\\\\\boxed{sin\alpha=\dfrac{2\sqrt6}5}[/tex]
[tex]tg\alpha=\dfrac{2\sqrt6}{5\!\!\!\!\diagup_1}*(-5\!\!\!\!\diagup^1)\\\\\boxed{tg\alpha=-2\sqrt6}[/tex]
[tex]ctg\alpha=-\dfrac{1}{2\sqrt6}*\dfrac{\sqrt6}{\sqrt6}\\\\\boxed{ctg\alpha=-\dfrac{\sqrt6}{12}}[/tex]
c)
[tex]tg\alpha=-1\dfrac14; sin\alpha > 0; cos\alpha, ctg\alpha < 0\\\\-1\dfrac14=\dfrac{sin\alpha}{cos\alpha} /^2\\\\\left(-\dfrac54\right)^2=\dfrac{sin^2\alpha}{cos^2\alpha}\\\\\dfrac{25}{16}=\dfrac{sin^2\alpha}{1-sin^2\alpha}\\\\\\25(1-sin^2\alpha)=16sin^2\alpha\\25-25sin^2\alpha=16sin^2\alpha\\-25sin^2\alpha-16sin^2\alpha=-25\\-41sin^2\alpha=-25 /:(-41)\\sin^2\alpha=\dfrac{25}{41}\\\\sin\alpha=\dfrac{5}{\sqrt{41}} /*\dfrac{\sqrt{41}}{\sqrt{41}}\\\\\boxed{sin\alpha=\dfrac{5\sqrt{41}}{41}}[/tex]
[tex]\dfrac{25}{41}+cos^2\alpha=1 /-\dfrac{25}{41}\\\\cos^2\alpha=\dfrac{16}{41}\\\\cos\alpha=-\dfrac4{\sqrt{41}}*\dfrac{\sqrt{41}}{\sqrt{41}}\\\\\boxed{cos\alpha=-\dfrac{4\sqrt{41}}{41}}[/tex]
[tex]ctg\alpha=1:\left(-\dfrac54\right)\\\\ctg\alpha=1*\left(-\dfrac45\right)\\\\\boxed{ctg\alpha=-\dfrac45}[/tex]
