Question

Blake is told that an angle has a cosine of −16√
and is in Quadrant III. Use the Pythagorean identity sin2(θ)+cos2(θ)=1
, the trigonometric identity tanθ=sinθcosθ
, and the quadrant to find the tangent.

Answer 1

It seems like there might be a typo in the cosine value you provided, as the cosine of an angle cannot be greater than 1 in absolute value. Assuming you meant the cosine to be \(-\frac{16}{\sqrt{17}}\) (since \(-16\sqrt{17}\) is not feasible), I will proceed with that assumption.

Given:
\[
\cos(\theta) = -\frac{16}{\sqrt{17}}
\]

### Step 1: Find \(\sin(\theta)\) using the Pythagorean identity
The Pythagorean identity tells us that:
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
Substituting the value of \(\cos(\theta)\):
\[
\sin^2(\theta) + \left(-\frac{16}{\sqrt{17}}\right)^2 = 1
\]
Calculating \(\cos^2(\theta)\):
\[
\sin^2(\theta) + \frac{256}{17} = 1
\]
Rearranging gives:
\[
\sin^2(\theta) = 1 - \frac{256}{17} = \frac{17}{17} - \frac{256}{17} = \frac{-239}{17}
\]
This is incorrect, as the sine squared cannot be negative. Thus the assumption about the cosine might be completely off.

### Consider an alternate approach
Visitors often provide ratios instead of exact values. If you confirm or provide the right ratio for \(\cos(\theta)\), for example, \(-\frac{3}{5}\) or any proper trigonometric ratio, then we can proceed with calculations based solely on that known quantity.

### Step 2: Determine \(\tan(\theta)\)
The tangent identity states:
\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\]
So after finding \(\sin(\theta)\), you can find \(\tan(\theta)\) accordingly.

Please clarify the cosine value so we can continue with accurate calculations!