Question

1. use the unit circle to find the value of tan315°. 20. find the amplitude and period of the function f(x)= - 2sinx.

Explanation:

Step1: Recall tangent - function formula

The formula for the tangent function is $\tan\theta=\frac{\sin\theta}{\cos\theta}$. For an angle $\theta$ on the unit - circle, if the coordinates of the corresponding point are $(x,y)$, then $\cos\theta = x$ and $\sin\theta=y$.

Step2: Identify coordinates for $315^{\circ}$

From the unit - circle, for $\theta = 315^{\circ}$, the coordinates of the corresponding point are $(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})$. So, $\cos315^{\circ}=\frac{\sqrt{2}}{2}$ and $\sin315^{\circ}=-\frac{\sqrt{2}}{2}$.

Step3: Calculate $\tan315^{\circ}$

Using the formula $\tan315^{\circ}=\frac{\sin315^{\circ}}{\cos315^{\circ}}$, substitute $\sin315^{\circ}=-\frac{\sqrt{2}}{2}$ and $\cos315^{\circ}=\frac{\sqrt{2}}{2}$ into it. Then $\tan315^{\circ}=\frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}=- 1$.

Answer:

$-1$