find the exact value of each of the six trigonometric functions of theta, if (6, 2) is a point on the terminal side of angle theta.

We are given a point \((6, 2)\) on the terminal side of an angle \(\theta\) in standard position. To find the six trigonometric functions of \(\theta\), we first need to determine the radius \(r\), which is the distance from the origin to the point \((6, 2)\). This radius is given by the Pythagorean theorem:

\[
r = \sqrt{x^2 + y^2} = \sqrt{6^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10}.
\]

Now, we can calculate the six trigonometric functions of \(\theta\):

1. **Sine** \(\sin \theta\):
\[
\sin \theta = \frac{y}{r} = \frac{2}{2\sqrt{10}} = \frac{1}{\sqrt{10}} = \frac{\sqrt{10}}{10}.
\]

2. **Cosine** \(\cos \theta\):
\[
\cos \theta = \frac{x}{r} = \frac{6}{2\sqrt{10}} = \frac{3}{\sqrt{10}} = \frac{3\sqrt{10}}{10}.
\]

3. **Tangent** \(\tan \theta\):
\[
\tan \theta = \frac{y}{x} = \frac{2}{6} = \frac{1}{3}.
\]

4. **Cosecant** \(\csc \theta\) (the reciprocal of sine):
\[
\csc \theta = \frac{r}{y} = \frac{2\sqrt{10}}{2} = \sqrt{10}.
\]

5. **Secant** \(\sec \theta\) (the reciprocal of cosine):
\[
\sec \theta = \frac{r}{x} = \frac{2\sqrt{10}}{6} = \frac{\sqrt{10}}{3}.
\]

6. **Cotangent** \(\cot \theta\) (the reciprocal of tangent):
\[
\cot \theta = \frac{x}{y} = \frac{6}{2} = 3.
\]

Thus, the exact values of the six trigonometric functions are:

- \(\sin \theta = \frac{\sqrt{10}}{10}\),
- \(\cos \theta = \frac{3\sqrt{10}}{10}\),
- \(\tan \theta = \frac{1}{3}\),
- \(\csc \theta = \sqrt{10}\),
- \(\sec \theta = \frac{\sqrt{10}}{3}\),
- \(\cot \theta = 3\).

Would you like any further details or clarification on any of these steps?

Here are 5 related questions to explore:
1. How would these values change if the point were in a different quadrant?
2. Can you derive these trigonometric functions using unit circle principles?
3. How can the Pythagorean identity help verify these results?
4. What are the implications of negative angles in this context?
5. How do trigonometric functions relate to real-world applications like navigation?

**Tip**: The reciprocal identities are a quick way to check your work when finding cosecant, secant, and cotangent!

Find the exact value of each of the six trigonometric functions of theta, if (6, 2) is a point on the terminal side of angle theta.

Find the exact values of the six trigonometric functions for angle theta, given the point (6, 2) on the terminal side. This step-by-step solution uses the Pythagorean theorem to calculate sine, cosine, tangent, and their reciprocal functions. Perfect for students in Grades 9-12.

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