Use identities to find the exact value of each of the four remaining trigonometric functions of the acute angle theta.
sine theta equals two sevenths comma cosine theta equals StartFraction 3 StartRoot 5 EndRoot Over 7 EndFraction
Question content area bottom
Part 1
tangent theta equals nothing

​(Simplify your​ answer, including any radicals. Use integers or fractions for any numbers in the​ expression.)

To find the remaining trigonometric functions of the acute angle \(\theta\), we have:

- \(\sin \theta = \frac{2}{7}\)
- \(\cos \theta = \frac{3 \sqrt{5}}{7}\)

We will use the trigonometric identities and the values of \(\sin \theta\) and \(\cos \theta\) to determine \(\tan \theta\), \(\cot \theta\), \(\sec \theta\), and \(\csc \theta\).

### Step 1: Find \(\tan \theta\)
The tangent of an angle is given by:
\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\]

Substitute the values of \(\sin \theta\) and \(\cos \theta\):
\[
\tan \theta = \frac{\frac{2}{7}}{\frac{3 \sqrt{5}}{7}} = \frac{2}{3 \sqrt{5}}
\]

To rationalize the denominator:
\[
\tan \theta = \frac{2 \cdot \sqrt{5}}{3 \cdot 5} = \frac{2 \sqrt{5}}{15}
\]

Thus:
\[
\tan \theta = \frac{2 \sqrt{5}}{15}
\]

### Step 2: Find \(\cot \theta\)
The cotangent is the reciprocal of the tangent:
\[
\cot \theta = \frac{1}{\tan \theta} = \frac{15}{2 \sqrt{5}}
\]

Rationalizing the denominator:
\[
\cot \theta = \frac{15 \cdot \sqrt{5}}{2 \cdot 5} = \frac{3 \sqrt{5}}{2}
\]

So:
\[
\cot \theta = \frac{3 \sqrt{5}}{2}
\]

### Step 3: Find \(\sec \theta\)
The secant is the reciprocal of the cosine:
\[
\sec \theta = \frac{1}{\cos \theta} = \frac{7}{3 \sqrt{5}}
\]

Rationalizing the denominator:
\[
\sec \theta = \frac{7 \cdot \sqrt{5}}{3 \cdot 5} = \frac{7 \sqrt{5}}{15}
\]

So:
\[
\sec \theta = \frac{7 \sqrt{5}}{15}
\]

### Step 4: Find \(\csc \theta\)
The cosecant is the reciprocal of the sine:
\[
\csc \theta = \frac{1}{\sin \theta} = \frac{7}{2}
\]

Thus:
\[
\csc \theta = \frac{7}{2}
\]

### Summary of Results
- \(\tan \theta = \frac{2 \sqrt{5}}{15}\)
- \(\cot \theta = \frac{3 \sqrt{5}}{2}\)
- \(\sec \theta = \frac{7 \sqrt{5}}{15}\)
- \(\csc \theta = \frac{7}{2}\)

Would you like further clarification on any of these calculations?

Here are 5 related questions:
1. How can these trigonometric identities be used in other applications?
2. What if \(\theta\) were in a different quadrant? How would the signs change?
3. How do the Pythagorean identities relate to these trigonometric functions?
4. What is the significance of rationalizing the denominator?
5. How can exact values like these be used in real-life trigonometric applications?

**Tip:** Always remember to rationalize denominators when expressing trigonometric functions for exact answers!

Use identities to find the exact value of each of the four remaining trigonometric functions of the acute angle theta. sine theta equals two sevenths, cosine theta equals 3√5/7. Tangent theta equals nothing.

Learn how to find the exact values of tangent, cotangent, secant, and cosecant using trigonometric identities. Given sin(θ) = 2/7 and cos(θ) = 3√5/7, this solution covers key formulas and rationalizing denominators. Ideal for high school students studying trigonometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation