To find the values of the six trigonometric functions of the angle \(\theta\) whose terminal side lies on the given line \( y = 6x, \, x \geq 0 \), we follow these steps:
1. Select a Point on the Line: The line \( y = 6x \) passes through the origin, and any point \((x, y)\) where \( y = 6x \) lies on this line. For simplicity, we can choose \( x = 1 \).
[tex]\[
x = 1, \quad y = 6 \times 1 = 6
\][/tex]
2. Calculate the Hypotenuse \(r\): Use the Pythagorean theorem to find \( r \), the length of the hypotenuse of the right triangle formed by \( x \), \( y \), and \( r \).
[tex]\[
r = \sqrt{x^2 + y^2} = \sqrt{1^2 + 6^2} = \sqrt{1 + 36} = \sqrt{37}
\][/tex]
3. Find the Six Trigonometric Functions:
- Sine (\(\sin \theta\)):
[tex]\[
\sin \theta = \frac{y}{r} = \frac{6}{\sqrt{37}}
\][/tex]
Rationalize the denominator:
[tex]\[
\sin \theta = \frac{6}{\sqrt{37}} \cdot \frac{\sqrt{37}}{\sqrt{37}} = \frac{6 \sqrt{37}}{37}
\][/tex]
- Cosine (\(\cos \theta\)):
[tex]\[
\cos \theta = \frac{x}{r} = \frac{1}{\sqrt{37}}
\][/tex]
Rationalize the denominator:
[tex]\[
\cos \theta = \frac{1}{\sqrt{37}} \cdot \frac{\sqrt{37}}{\sqrt{37}} = \frac{\sqrt{37}}{37}
\][/tex]
- Tangent (\(\tan \theta\)):
[tex]\[
\tan \theta = \frac{y}{x} = \frac{6}{1} = 6
\][/tex]
- Secant (\(\sec \theta\)):
[tex]\[
\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{\sqrt{37}}{37}} = \frac{37}{\sqrt{37}} = \sqrt{37}
\][/tex]
Rationalize the denominator:
[tex]\[
\sec \theta = \sqrt{37}
\][/tex]
- Cosecant (\(\csc \theta\)):
[tex]\[
\csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{6 \sqrt{37}}{37}} = \frac{37}{6 \sqrt{37}} = \frac{37}{6 \sqrt{37}}
\][/tex]
Rationalize the denominator:
[tex]\[
\csc \theta = \frac{37 \sqrt{37}}{6 \cdot 37} = \frac{\sqrt{37}}{6}
\][/tex]
- Cotangent (\(\cot \theta\)):
[tex]\[
\cot \theta = \frac{1}{\tan \theta} = \frac{1}{6}
\][/tex]
Therefore, the exact, fully simplified values of the six trigonometric functions are:
[tex]\[
\sin \theta = \frac{6 \sqrt{37}}{37}, \;\; \cos \theta = \frac{\sqrt{37}}{37}, \;\; \tan \theta = 6, \;\; \sec \theta = \sqrt{37}, \;\; \csc \theta = \frac{\sqrt{37}}{6}, \;\; \cot \theta = \frac{1}{6}
\][/tex]
