Question

What is the tangent for an angle that has a cosine of −17√ and is in Quadrant III? Use the Pythagorean identity sin2θ+cos2θ=1 , the trigonometric identity tanθ=sinθcosθ , and the quadrant to solve. (1 point) Responses 6 6 6–√ Start Root 6 End Root −6 negative 6 −6–√

Answer 1

To find the tangent of the angle with a cosine of \(-\sqrt{17}\) in Quadrant III, we first need to determine the sine of the angle using the Pythagorean identity:

\[
\sin^2 \theta + \cos^2 \theta = 1.
\]

Given that \(\cos \theta = -\sqrt{17}\):

\[
\sin^2 \theta + (-\sqrt{17})^2 = 1.
\]

Calculating \((- \sqrt{17})^2\):

\[
\sin^2 \theta + 17 = 1.
\]

Now, rearranging to find \(\sin^2 \theta\):

\[
\sin^2 \theta = 1 - 17 = -16.
\]

This results in a negative value, which indicates a mistake in the setup since the cosine value should represent a valid real number. Since cosines generally lie between -1 and 1 for angles on the unit circle, let's clarify that \(\cos \theta = -\sqrt{17}\) is not a valid real cosine value.

Correcting this based on the usual context:

If we assume the given cosine was incorrectly noted, say we have:

For example, with \(cos(\theta) = -\frac{4}{5}\) or any value \(< 1\):

1. Confirming with a valid \(\cos\) from generally real valued identities we could continue:

Continue solving the angle for sin value, then tangent from there.

**Without any clear numerical value, solution cannot be derived using given cosine of \(-\sqrt{17}\).**

Let's try reworking with an illustrative example close to valid:

Assuming cosine might instead be \(-\frac{3}{5}\):

1. \(\sin^2 \theta + (-\frac{3}{5})^2 = 1 \implies sin^2 \theta + \frac{9}{25} = 1 \implies sin^2 \theta + \frac{9}{25} = \frac{25}{25} \implies sin^2 \theta = \frac{16}{25} \implies sin \theta = -\frac{4}{5}\)

Using this:

Then, \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{4}{5}}{-\frac{3}{5}} = \frac{4}{3}\).

If we calculate your original correctly, ensure the range and values are valid as analytical interpretation states.

For valid cosine/real computed angle, maintain fitness with note checking - clarify the angle setups!

**Therefore**, under reexamination the answer may not yield from negative square roots ascertainable through standard trigonometric identities. Review context of original prompt check on those values! ![So to conclude values can’t support ascertainable tangent, thus look confirmatively check back on conditions].