Question

macy is told that an angle has a sine of √(not square root symbol just the checkmark)5/5 and is in quadrant 2 find cosine and tangent using identities and the quadrant

Answer 1

To find the cosine and tangent of an angle when given the sine, we can use trigonometric identities, particularly the Pythagorean identity, which states:

\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]

Given that \(\sin(\theta) = \frac{\check{5}}{5}\), we can start by calculating \(\cos(\theta)\).

1. **Calculate \(\cos(\theta)\)**:

First, we square the sine value:

\[
\sin^2(\theta) = \left(\frac{\check{5}}{5}\right)^2 = \frac{5}{25} = \frac{1}{5}
\]

Now, use the Pythagorean identity:

\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]

Substituting the value of \(\sin^2(\theta)\):

\[
\frac{1}{5} + \cos^2(\theta) = 1
\]

To find \(\cos^2(\theta)\), subtract \(\frac{1}{5}\) from both sides:

\[
\cos^2(\theta) = 1 - \frac{1}{5} = \frac{5}{5} - \frac{1}{5} = \frac{4}{5}
\]

Now, take the square root to find \(\cos(\theta)\). Since the angle is in quadrant 2, the cosine is negative:

\[
\cos(\theta) = -\sqrt{\frac{4}{5}} = -\frac{2}{\sqrt{5}} = -\frac{2\sqrt{5}}{5}
\]

2. **Calculate \(\tan(\theta)\)**:

Now that we have both sine and cosine, we can find the tangent:

\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\]

Substituting the values of \(\sin(\theta)\) and \(\cos(\theta)\):

\[
\tan(\theta) = \frac{\frac{\check{5}}{5}}{-\frac{2\sqrt{5}}{5}} = \frac{\check{5}}{-2\sqrt{5}} = -\frac{1}{2}
\]

Thus, the cosine and tangent of the angle are:

\[
\cos(\theta) = -\frac{2\sqrt{5}}{5}
\]
\[
\tan(\theta) = -\frac{1}{2}
\]