Question 6074d

May 2, 2017

see explanation below

Explanation:

in quadrant 2, the value of x-axis is -ve while y-axis is +ve and it hypotenuse is +ve value.

sin = y-axis/hypotenuse = +ve value.

cosec = 1/sin = hypotenuse/y-axis = +ve value.

the rest are involve with the #x-axis and become -ve value.

May 2, 2017

It has to do with the signs of the x and y axis in the second quadrant.

We let $\sin \theta = \frac{b}{r}$, $\cos \theta = \frac{a}{r}$ and $\tan \theta = \frac{b}{a}$, where ${a}^{2} + {b}^{2} = {r}^{2}$. Here's why: The green triangle is right because side $b$ is parallel to the y-axis and the x and y axis are perpendicular to each other. The hypotenuse will always be positive, so it's $a$ and $b$ that will influence the sign of the ratio.

We have $\sin \theta = \text{opposite"/"hypotenuse} = \frac{b}{r}$ and $b$ will be positive because the y-axis in quadrant II is positive. Therefore, sine will always be positive in quadrant $I I$. This also applies to cosecant because $\csc x = \frac{1}{\sin} x$.

Hopefully this helps!