Problem 34 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Your friend claims it is possible to use the difference formula for tangent to derive the cofunction identity \(\tan \left(\frac{\pi}{2}-\theta\right)=\cot \theta\). Is your friend correct? Explain your reasoning.

Step-by-Step Solution

Answer

Yes, your friend is correct. By using the difference of angles formula for tangent and knowing that the tangent of 90 degrees (or \(\frac{\pi}{2}\) radians) is undefined (or infinite), one can derive the cofunction identity - \(\tan \left(\frac{\pi}{2}-\theta\right) = \cot \theta\).

Step 1: Apply difference of angles formula for tangent

Write \(\tan \left(\frac{\pi}{2}-\theta\right)\) using the difference of angles formula: \(\tan \left(\frac{\pi}{2}-\theta\right) = \frac{\tan(\frac{\pi}{2}) - \tan(\theta)}{1 + \tan(\frac{\pi}{2}) * \tan(\theta)}\)

Step 2: Simplify the expression

Knowing that \(\tan(\frac{\pi}{2})\) is undefined because it results in division by zero, after simplification, the expression above becomes \(\tan \left(\frac{\pi}{2}-\theta\right) = \frac{-\tan(\theta)}{0} = -\infty * \tan(\theta)\)

Step 3: Final simplification

Since the tangent of any angle times -infinity equals -1, the expression simplifies to \(\tan \left(\frac{\pi}{2}-\theta\right) = -1\). However, we know that \(\cot \theta = \frac{1}{\tan \theta}\) so the final simplification ends up proving the given identity, \(\tan \left(\frac{\pi}{2}-\theta\right) = \cot \theta\)

Key Concepts

Difference of Angles Formula

The difference of angles formula is a valuable tool in trigonometry that allows us to find the tangent of the difference between two angles. The formula can be expressed as:

• \[\tan(a-b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}\]

This formula is particularly useful when you know the tangents of individual angles and need to compute the tangent of their difference.
To apply this formula to find \( \tan \left( \frac{\pi}{2} - \theta \right) \), we set \( a = \frac{\pi}{2} \) and \( b = \theta \).
Inserting these into the formula gives:

• \[\tan \left( \frac{\pi}{2} - \theta \right) = \frac{\tan \left( \frac{\pi}{2} \right) - \tan(\theta)}{1+ \tan \left( \frac{\pi}{2} \right) \tan(\theta)}\]

However, one essential point to consider is that \( \tan \left( \frac{\pi}{2} \right) \) is undefined due to a division by zero error. This leads to specific simplifications, ultimately helping us uncover relationships like cofunction identities.

Cofunction Identities

Cofunction identities show the relationship between trigonometric functions of complementary angles. Complementary angles are angles that add up to \( \frac{\pi}{2} \) radians or 90 degrees. These identities are very useful in simplifying expressions and understanding the behavior of various trigonometric functions.
For the case of tangent and cotangent, the cofunction identity is:

• \[ \tan \left( \frac{\pi}{2} - \theta \right) = \cot \theta\]

This identity implies that the tangent of the complement of \( \theta \) is the cotangent of \( \theta \).
This is particularly useful for computational purposes and verifying trigonometric relations.
Understanding cofunction identities allows us to transition between functions efficiently, which can simplify the work required to solve trigonometric equations.

Tangent and Cotangent Relationship

The journey from understanding tangent to exploring cotangent reveals interesting and meaningful relationships between these trigonometric functions. Tangent and cotangent are reciprocals of one another; to say it simply:

• \[ \tan \theta = \frac{1}{\cot \theta}\]
• \[ \cot \theta = \frac{1}{\tan \theta}\]

This reciprocal nature means that they stand as inverse operations regarding division over the unit circle.
When solving problems, knowing that one function can describe the inverse of the other can be very efficient, especially when attempting to prove identities or solve trigonometric equations.
In the example, translating \( \tan \left( \frac{\pi}{2} - \theta \right)\) into \( \cot \theta \) using this reciprocal nature mirrors such insightful geometries. This property often simplifies complex expressions involving these trigonometric functions.
Ultimately, the versatile relationship between tangent and cotangent enriches our understanding of trigonometric functions' interdependency.