Question

Write the first expression in terms of the second if the terminal point determined by $t$is in the given quadrant.

$\sin t,\cos t;\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Quadrant\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}II}$

Short answer

The first expression in terms of the second if the terminal point determined by $t$is in the given quadrant is$\mathbf{sin}\mathit{t}\mathbf{=}\sqrt{\mathbf{1}\mathbf{-}{\mathbf{cos}}^{\mathbf{2}}\mathbf{t}}$.

Step-by-step solution

Step 1. State the Pythagorean identities.

The Pythagorean identities are:

1. ${\sin }^{2}t+{\cos }^{2}t=1$

2. ${\tan }^{2}t+1={\sec }^{2}t$

3.${\cot }^{2}t+1={\csc }^{2}t$

Step 2. Rewrite the expression in terms of cost. 

Using Pythagorean Identity ${\sin }^{2}t+{\cos }^{2}t=1$, rewrite the expression $\sin t$into terms of $\cos t$.

$$\begin{gathered} {\sin }^{2}t+{\cos }^{2}t=1 \\ {\sin }^{2}t=1-{\cos }^{2}t\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Subtract\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\cos }^{2}t\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}onbothsides} \\ \sin t=\pm \sqrt{1-{\cos }^{2}t}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Takesquarerootbothsides} \end{gathered}$$

Step 3. State the conclusion.

Now, $t$ is given to be in quadrant II, and quadrant II, $\sin t>0$. Therefore, $\sin t=-\sqrt{1-{\cos }^{2}t}$ is not a valid expression. So, $\sin t=\sqrt{1-{\cos }^{2}t}$.