Evaluate all trig functions for secant of theta=7/3

Given that \(\sec(\theta) = \frac{7}{3}\), we can find all the trigonometric functions step by step.

### 1. Cosine
The secant function is the reciprocal of the cosine function:

\[
\cos(\theta) = \frac{1}{\sec(\theta)} = \frac{1}{\frac{7}{3}} = \frac{3}{7}
\]

### 2. Sine
To find \(\sin(\theta)\), we can use the Pythagorean identity:

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

Substituting \(\cos(\theta) = \frac{3}{7}\):

\[
\sin^2(\theta) + \left(\frac{3}{7}\right)^2 = 1
\]

\[
\sin^2(\theta) + \frac{9}{49} = 1
\]

\[
\sin^2(\theta) = 1 - \frac{9}{49} = \frac{49}{49} - \frac{9}{49} = \frac{40}{49}
\]

\[
\sin(\theta) = \sqrt{\frac{40}{49}} = \frac{\sqrt{40}}{7} = \frac{2\sqrt{10}}{7}
\]

### 3. Tangent
The tangent function is the ratio of sine to cosine:

\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{2\sqrt{10}}{7}}{\frac{3}{7}} = \frac{2\sqrt{10}}{3}
\]

### 4. Cosecant
The cosecant function is the reciprocal of the sine function:

\[
\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{7}{2\sqrt{10}} = \frac{7\sqrt{10}}{20} \quad \text{(rationalized)}
\]

### 5. Cotangent
The cotangent function is the reciprocal of the tangent function:

\[
\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{3}{2\sqrt{10}} = \frac{3\sqrt{10}}{20} \quad \text{(rationalized)}
\]

So, all the trigonometric functions are:

- \(\sec(\theta) = \frac{7}{3}\)
- \(\cos(\theta) = \frac{3}{7}\)
- \(\sin(\theta) = \frac{2\sqrt{10}}{7}\)
- \(\tan(\theta) = \frac{2\sqrt{10}}{3}\)
- \(\csc(\theta) = \frac{7\sqrt{10}}{20}\)
- \(\cot(\theta) = \frac{3\sqrt{10}}{20}\)

Feel free to ask any more questions if needed!

Here are 5 related questions:
1. What is the angle \(\theta\) in degrees or radians given \(\sec(\theta) = \frac{7}{3}\)?
2. How do you find the inverse trigonometric functions given \(\sec(\theta) = \frac{7}{3}\)?
3. What is the significance of rationalizing denominators in trigonometric expressions?
4. How can you verify the results of trigonometric functions using a calculator?
5. How does knowing one trigonometric function help in finding all others?

**Tip:** When dealing with square roots in trigonometric identities, always consider both positive and negative roots depending on the quadrant of the angle \(\theta\).

Explore how to evaluate all trigonometric functions including cosine, sine, tangent, cosecant, and cotangent when secant theta equals 7 over 3. Detailed step-by-step solution provided.

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