sin of theta = -13/85, what is cos of theta?

To find \(\cos(\theta)\) given that \(\sin(\theta) = -\frac{13}{85}\), we can use the Pythagorean identity:

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

### Step 1: Square the sine value
\[
\sin^2(\theta) = \left(-\frac{13}{85}\right)^2 = \frac{169}{7225}
\]

### Step 2: Use the Pythagorean identity
\[
\cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \frac{169}{7225}
\]

To subtract these, we write \(1\) as \(\frac{7225}{7225}\):

\[
\cos^2(\theta) = \frac{7225}{7225} - \frac{169}{7225} = \frac{7056}{7225}
\]

### Step 3: Take the square root
\[
\cos(\theta) = \pm \sqrt{\frac{7056}{7225}} = \pm \frac{\sqrt{7056}}{85}
\]

The square root of 7056 is 84:

\[
\cos(\theta) = \pm \frac{84}{85}
\]

### Step 4: Determine the sign of \(\cos(\theta)\)
Since \(\sin(\theta)\) is negative, and \(\sin(\theta)\) is negative in the third and fourth quadrants, \(\cos(\theta)\) would be negative in the third quadrant and positive in the fourth quadrant. Without additional information about the angle \(\theta\), both values are possible, so:

\[
\cos(\theta) = \pm \frac{84}{85}
\]

**Note:** If you know the quadrant \(\theta\) lies in, you can determine the exact sign.

Would you like more details, or do you have any further questions?

### Related Questions:
1. How do you determine the quadrant of an angle given sine and cosine?
2. How would the problem change if \(\sin(\theta)\) were positive?
3. What is the tangent of \(\theta\) given this information?
4. How does the Pythagorean identity help in solving trigonometric equations?
5. What are the possible angles for \(\theta\) given this sine value?

### Tip:
Always remember the signs of trigonometric functions in each quadrant: "All Students Take Calculus" (All, Sine, Tangent, Cosine) helps to recall where each function is positive.

Learn how to find the cosine of theta given the sine of theta equals negative 13 over 85. Step-by-step solution using the Pythagorean identity and trigonometric principles. Suitable for high school students learning trigonometry.

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