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== [[Funkcija (matematika)|Funkcije]] jednog [[Kut|kuta]] ==
: <math>\sin ^2\alpha + \cos ^2 \alpha = 1, \quad \frac{\sin \alpha}{\cos \alpha}=\tan \alpha, \quad \sin \alpha \cdot \csc \alpha = 1</math>,
: <math>\sec ^2 \alpha - \tan ^2 \alpha = 1, \qquad \cos \alpha \cdot \sec \alpha = 1</math>,
: <math>\csc ^2 \alpha - \cot ^2 \alpha = 1, \quad \frac{\cos \alpha}{\sin \alpha} = \cot \alpha, \quad \tan \alpha \cdot \cot \alpha = 1</math>

== Međusobno izražavanje funkcija ==
: <math> \sin \alpha = \sqrt{1 - \cos ^2 \alpha} = \frac{ \tan \alpha}{ \sqrt{ 1 + \tan ^2 \alpha}},</math>
: <math>\cos \alpha = \sqrt{1- \sin ^2 \alpha}=\frac{1}{\sqrt{1+ \tan ^2 \alpha}} ,</math>
: <math>\tan \alpha = \frac{\sin \alpha}{\sqrt{1- \sin ^2\alpha}}=\frac{1}{\cot \alpha},</math>
: <math>\cot \alpha = \frac{\sqrt{1- \sin ^2\alpha}}{\sin \alpha}= \frac{1}{\tan \alpha}.</math>

== Funkcije zbroja i razlike ==
: <math>\sin ( \alpha \pm \beta )= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta,\,</math>
: <math>\cos (\alpha \pm \beta )= \cos \alpha \cos \beta \mp \sin \alpha \sin \beta,</math>
: <math>\tan (\alpha \pm \beta )=\frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}, \quad \cot ( \alpha \pm \beta ) = \frac{\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha}.</math>
: <math>\tan2\alpha=\frac{2\tan\alpha}{1-\tan^2\alpha}, \tan3\alpha=\frac{3\tan\alpha-\tan^3\alpha}{1-3\tan^2\alpha},</math>

: <math>\sin2\alpha = 2\sin\alpha\cos\alpha, \quad \sin3\alpha=3\sin\alpha-4\sin^3\alpha,</math>
: <math>\cos2\alpha = \cos^2\alpha-\sin^2\alpha, \quad \cos3\alpha=4\cos^3\alpha-3\cos\alpha,</math>
: <math>\tan2\alpha=\frac{2\tan\alpha}{1-\tan^2\alpha}, \quad \tan3\alpha=\frac{3\tan\alpha-\tan^3\alpha}{1-3\tan^2\alpha},</math>
: <math>\cot2\alpha=\frac{\cot^2\alpha-1}{2\cot\alpha}, \quad \cot3\alpha=\frac{\cot^3\alpha-3\cot\alpha}{3\cot^2\alpha-1},</math>
: <math>\tan4\alpha=\frac{4\tan\alpha-4\tan^3\alpha}{1-6\tan^2\alpha+\tan^4\alpha}, \quad \cot4\alpha=\frac{\cot^4\alpha-6\cot^2\alpha+1}{4\cot^3\alpha-4\cot\alpha}.</math>
Na osnovu ovih formula možemo odrediti predznak trigonometrijskih funkcija po kvadrantima
{| align="center" border="1" cellpadding="2" style="margin-bottom: 10px;"
! Kut
! 0°- 90°
! 90°- 180°
! 180°- 270°
! 270°- 360°
|-
!Kvadrant
!I.
!II.
!III.
!IV.
|- align="center"
! sinus
| +
| +
| -
| -
|- align="center"
! kosinus
| +
| -
| -
| +
|- align="center"
! tangens
| +
| -
| +
| -
|-
|-align="center"
!kotangens
| +
| -
| -
| +
|}

== Zbroj i razlika trigonometrijskih funkcija ==
: <math>\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2},</math>
: <math>\sin\alpha-\sin\beta=2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2},</math>
: <math>\cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2},</math>
: <math>\cos\alpha-\cos\beta=-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2},</math>
: <math>\tan\alpha\pm\tan\beta=\frac{\sin (\alpha\pm\beta )}{\cos\alpha\cos\beta}, \quad \cot\alpha\pm\cot\beta=\pm\frac{\sin (\alpha\pm\beta)}{\sin\alpha\sin\beta},</math>
: <math>\tan\alpha+\cot\beta=\frac{\cos (\alpha-\beta)}{\cos\alpha\sin\beta}, \quad \cot\alpha-\tan\beta=\frac{cos (\alpha+\beta)}{\sin\alpha\cos\beta}.</math>

== Umnožak funkcija ==
: <math>\sin\alpha\sin\beta=\frac{1}{2}[\cos(\alpha-\beta)-\cos(\alpha+\beta)],</math>
: <math>\cos\alpha\cos\beta=\frac{1}{2}[\cos(\alpha-\beta)+cos(\alpha+\beta)],</math>
: <math>\sin\alpha\cos\beta=\frac{1}{2}[\sin(\alpha+\beta)+\sin(\alpha-\beta)].</math>

== Funkcije polovine kuta ==
: <math>\sin\frac{\alpha}{2}=\sqrt{\frac{1-\cos\alpha}{2}}, \quad \cos\frac{\alpha}{2}=\sqrt{\frac{1+\cos\alpha}{2}},</math>
: <math>\tan\frac{\alpha}{2}=\sqrt{\frac{1-\cos\alpha}{1+\cos\alpha}}=\frac{1-\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1+\cos\alpha},</math>
: <math>\cot\frac{\alpha}{2}=\sqrt{\frac{1+\cos\alpha}{1-\cos\alpha}}=\frac{1+\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1-\cos\alpha}.</math>

== Potenciranje funkcija ==
<math>\sin^2\alpha=\frac{1}{2}(1-\cos2\alpha), \quad \cos^2\alpha=\frac{1}{2}(1+\cos2\alpha),</math>

<math>\sin^3\alpha=\frac{1}{4}(3\sin\alpha-\sin3\alpha), \quad \cos^3\alpha=\frac{1}{4}(\cos3\alpha+3\cos\alpha),</math>
<math>\sin^4\alpha=\frac{1}{8}(\cos4\alpha-4\cos2\alpha+3), \quad \cos^4\alpha=\frac{1}{8}(\cos4\alpha+4\cos2\alpha+3).</math>

[[Kategorija:Trigonometrija]]
[[Kategorija:Trigonometrijske funkcije]]

Osnovne trigonometrijske formule - Povijest promjena

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