Problem 54 - Algebra and Trigonometry

Question

\(51-54\). Evaluate each expression under the given conditions. $$ \tan 2 \theta ; \cos \theta=\frac{3}{5}, \theta \text { in Quadrant I } $$

Step-by-Step Solution

Answer

\(-\frac{24}{7}\).

Step 1: Identify Given Information

We are given that \(\cos \theta = \frac{3}{5}\) and \(\theta\) is in Quadrant I. We need to find \(\tan 2\theta\).

Step 2: Use the Pythagorean Identity

In Quadrant I, all trigonometric functions are positive. Use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \(\sin \theta\). Substitute \(\cos \theta = \frac{3}{5}\) into the identity: \(\sin^2 \theta = 1 - \left( \frac{3}{5} \right)^2 = 1 - \frac{9}{25} = \frac{16}{25}\). Therefore, \(\sin \theta = \frac{4}{5}\).

Step 3: Find \(\tan \theta\)

Using \(\sin \theta\) and \(\cos \theta\), calculate \(\tan \theta\) as: \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3}\).

Step 4: Apply Double Angle Formula for Tangent

Use the formula for double angle tangent: \(\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}\). Substitute \(\tan \theta = \frac{4}{3}\) into the formula to get: \(\tan 2\theta = \frac{2 \times \frac{4}{3}}{1 - \left( \frac{4}{3} \right)^2}\).

Step 5: Simplify \(\tan 2\theta\)

Calculate \(\tan^2 \theta = \left( \frac{4}{3} \right)^2 = \frac{16}{9}\). Substitute back into the double angle formula: \(\tan 2\theta = \frac{\frac{8}{3}}{1 - \frac{16}{9}} = \frac{\frac{8}{3}}{\frac{-7}{9}}\). Simplify the expression: \(\tan 2\theta = \frac{8}{3} \times \frac{9}{-7} = -\frac{24}{7}\).

Step 6: Conclusion

The value of \(\tan 2\theta\) is \(-\frac{24}{7}\).

Key Concepts

Double Angle Formulas

Double angle formulas are incredibly useful in trigonometry. They allow us to express trigonometric functions of double angles in terms of single angles. This is essential when we have a single angle and need to find trigonometric values for twice that angle. For the tangent function, the double angle formula is:

• \(\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}\)

This formula is derived from the sum of angles identity for tangent and simplifies many complex calculations. In this exercise, we use it to find \(\tan 2\theta\) when \(\tan \theta\) is already known. Understanding this formula helps in calculating the tangent of a double angle effortlessly.
This simplification allows us to apply an existing value into the equation and obtain the double angle's tangent quickly.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry. It relates the squares of sine and cosine of an angle. Formulated as \(\sin^2 \theta + \cos^2 \theta = 1\), this identity helps find missing trigonometric values when one is known. In the context of our exercise, knowing \(\cos \theta = \frac{3}{5}\), we use this identity to calculate \(\sin \theta\).

By rearranging the Pythagorean identity, we find:

• \(\sin^2 \theta = 1 - \cos^2 \theta\)
• Substitute \(\cos \theta = \frac{3}{5}\) to get: \(\sin^2 \theta = 1 - \frac{9}{25}\)
• Therefore, \(\sin^2 \theta = \frac{16}{25}\) and \(\sin \theta = \frac{4}{5}\)

Using this identity helps us unlock the rest of needed trigonometric values, ensuring all calculations are grounded within the same circle.

Tangent Function

The tangent function is a key component in trigonometry, defined as the ratio of sine to cosine. This function tells us how steep an angle is when compared to a circle's origin. When we know both \(\sin \theta\) and \(\cos \theta\), finding \(\tan \theta\) is straightforward:

• \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

In our case, using \(\sin \theta = \frac{4}{5}\) and \(\cos \theta = \frac{3}{5}\), we find:

• \(\tan \theta = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3}\)

This step is essential for later applying the double angle formula. Understanding the relationship between sine, cosine, and tangent gives a fuller picture of how each function interconnects with the others.

Trigonometric Functions in Quadrant I

Knowing the quadrant where an angle lies is crucial because it determines the signs of its trigonometric functions. In Quadrant I, all trigonometric functions—sine, cosine, and tangent—are positive. This agreement among the signs simplifies analyses.
For angles in Quadrant I:

• \(\sin \theta > 0\)
• \(\cos \theta > 0\)
• \(\tan \theta > 0\)

Since \(\theta\) is in Quadrant I for our problem, \(\cos \theta = \frac{3}{5}\) and \(\sin \theta = \frac{4}{5}\) both reflect positive values. This positivity confirms the signs when computing \(\tan \theta = \frac{4}{3}\). The quadrant information is vital as it aligns expected outcomes with mathematical results.