Problem 60 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 51-60: Show that the equation is not an identity. (Hint: Find one number for which the equation is false.) $$ \cot (\tan \theta)=1 $$

Step-by-Step Solution

Answer

The equation \( \cot(\tan \theta) = 1 \) is not an identity because it does not hold for \( \theta = \frac{\pi}{4} \).

Step 1: Understand the Equation

The given equation is \( \cot(\tan \theta) = 1 \). We need to determine if this is true for all \( \theta \). If it is not true for at least one value of \( \theta \), then the equation is not an identity.

Step 2: Test a Specific Value for \( \theta \)

Let's test \( \theta = \frac{\pi}{4} \). For this value, the tangent function gives \( \tan \left(\frac{\pi}{4}\right) = 1 \). Now, substitute this in the cotangent function.

Step 3: Calculate \( \cot(1) \)

\( \cot \theta \) is defined as \( \frac{1}{\tan \theta} \). So, \( \cot(1) = \frac{1}{1} eq 1 \) since \( 1 eq \frac{1}{1} \). We have shown \( \cot(1) eq 1 \) since \( \tan(1) = 1 \) was an assumption, and it does not hold for \( \tan(1) eq 1 \) in standard angle measurement.

Step 4: Conclusion

Since \( \cot(\tan(\theta)) eq 1 \) for \( \theta = \frac{\pi}{4} \), the equation \( \cot(\tan(\theta)) = 1 \) is not true universally. Hence, the equation does not hold as an identity.

Key Concepts

Equation Testing

Testing equations is an important process to understand if a statement applies to various values or specific conditions. In the context of trigonometric functions, it is often necessary to verify whether a given equation holds true for different values of the variable, \(\theta\). This process helps us determine if the equation is an identity.An identity is a statement that is true for all values within its domain. To show that a trigonometric equation is not an identity, we can test it with specific values of \(\theta\). If we find even one value where the equation does not hold, the equation is not universally true.In our exercise, we specifically tested \(\theta = \frac{\pi}{4}\). This very focused approach of verifying the equation worked well to demonstrate that the equation \(\cot(\tan(\theta)) = 1\) is not true for all values of \(\theta\). This approach can be applied to many mathematical problems involving equations.

Trigonometric Identity

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. These identities are fundamental tools in trigonometry because they allow the simplification and solving of complex expressions.Some classic examples of trigonometric identities include:

• \(\sin^2 \theta + \cos^2 \theta = 1\)
• \(1 + \tan^2 \theta = \sec^2 \theta\)
• \(1 + \cot^2 \theta = \csc^2 \theta\)

These identities serve as foundational truths that we can use in various calculations and proofs. However, not every equation that looks like it might be a trigonometric identity actually is one. For instance, the equation \(\cot(\tan(\theta)) = 1\) might superficially appear to be an identity, but upon testing, clearly does not hold for all \(\theta\). This example highlights the importance of understanding and verifying identities in mathematics.

Trigonometric Functions

Trigonometric functions are the building blocks of trigonometry. These functions relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions are sine, cosine, and tangent, often abbreviated as \(\sin\), \(\cos\), and \(\tan\). Complementary to these are the co-functions: cosecant (csc), secant (sec), and cotangent (cot).Each of these functions plays a crucial role in various mathematical contexts:

• \(\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}\)
• \(\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}\)
• \(\tan(\theta) = \sin(\theta)/\cos(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}\)
• \(\cot(\theta) = 1/\tan(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\)
• \(\sec(\theta) = 1/\cos(\theta)\)
• \(\csc(\theta) = 1/\sin(\theta)\)

These relationships can greatly aid in solving geometric problems and understanding wave behaviors. In our exercise, we tested functions such as tangent and cotangent to ascertain whether the initial given expression could be simplified universally. We found that it could not, demonstrating the potential complexities even seemingly simple equations can hold in trigonometry.