Chapter 7.5, Problem 1E

We can use identities to help us solve trigonometric equations.

1. Using a Pythagorean identity we see that the equation sin x + sin²x + cos²x = 1 is equivalent to the basic equation __________ whose solutions are x = _____.

To determine

To fill: The blanks in the statement “Using a Pythagorean identity we see that the equation $\sin x+{\sin }^{2}x+{\cos }^{2}x=1$ is equivalent to the basic equation ____________ whose solutions are x = ___________”.

Answer to Problem 1E

The complete statement is “Using a Pythagorean identity we see that the equation $\sin x+{\sin }^{2}x+{\cos }^{2}x=1$ is equivalent to the basic equation $\underset{¯}{\sin x=0}$ whose solutions are $\underset{¯}{x=n\pi }$”.

Explanation of Solution

Consider the given function $\sin x+{\sin }^{2}x+{\cos }^{2}x=1$.

Solve the above equation as follows.

$\begin{array}{l}\sin x+{\sin }^{2}x+{\cos }^{2}x=1\\ \sin x+1=1\\ \sin x=1-1\\ \sin x=0\end{array}$

Therefore, the basic equation is $\underset{¯}{\sin x=0}$.

Now, find the solutions of $\sin x=0$.

$\begin{array}{l}\sin x=0\\ x={\sin }^{-1}\left(0\right)\\ x=n\pi \end{array}$

Therefore, the solutions are $\underset{¯}{x=n\pi }$.