$\sin{\theta} \, > \, 0$
The sign of tan function in first quadrant of two-dimensional space is positive.
The sign of tan function in first quadrant of two-dimensional Cartesian coordinate system can be proved geometrically. It helps us to understand how the sign of tangent function in first quadrant is positive.
Thus, a right triangle is constructed geometrically and let’s take the angle of $\Delta EOF$ as theta ($\theta$). Therefore, the tangent function is written as $\tan{\theta}$ in trigonometry.
Let’s take the length of adjacent side is denoted by $x$ and the length of opposite side is represented by $y$. Now, express the tan function in ratio form of lengths of sides.
$\tan{\theta} \,=\, \dfrac{EF}{OF}$
In this case, $EF = y$ and $OF = x$.
$\implies \tan{\theta} \,=\, \dfrac{y}{x}$
In this case, the variable $x$ represents any value on the $x$-axis. Similarly, the variable $y$ represents any value on the $y$-axis. Actually, the $x$-axis and $y$-axis both represent positive values in first quadrant. Therefore, the lengths of opposite and adjacent sides should be positive. Therefore, $x > 0$ and $y > 0$.
$\implies \tan{\theta} \,=\, \dfrac{y}{x}$
In first quadrant, the values of $x$ and $y$ are positive. So, the quotient of them should also be positive. Therefore, the sign of tan function value is also positive.
$\,\,\, \therefore \,\,\,\,\,\, \tan{\theta} \, > \, 0$
$\Delta POQ$ is a right triangle in first quadrant of two dimensional space. The angle of this triangle is $\alpha$. The tangent function can be expressed in ratio form of lengths of sides.
$\tan{\alpha} \,=\, \dfrac{PQ}{OQ}$
In this example, $PQ \,=\, 4$ and $OQ \,=\, 6$. The sign of length of each side is positive. Now, substitute the lengths of sides in the ratio for calculating the tan value.
$\implies \,\,\,$ $\tan{\alpha} \,=\, \dfrac{4}{6}$
$\require{cancel} \implies \,\,\,$ $\tan{\alpha} \,=\, \dfrac{\cancel{4}}{\cancel{6}}$
$\implies \,\,\,$ $\tan{\alpha} \,=\, \dfrac{2}{3}$
The value of tan function $\tan{\alpha}$ is positive and it can be understood that the sign of tan function in first quadrant is positive.
A best free mathematics education website for students, teachers and researchers.
Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.
Learn how to solve the math problems in different methods with understandable steps and worksheets on every concept for your practice.
Copyright © 2012 - 2022 Math Doubts, All Rights Reserved