Problem 59 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use trigonometric identities to transform one side of the equation into the other \((0<{\theta}<\pi /2)\). $$\tan \theta \cot \theta=1$$

Step-by-Step Solution

Answer

After substituting the basic trigonometric identity \(\cot(\theta) = \frac{1}{\tan(\theta)}\), the original equation simplifies to \(1 = 1\), thus demonstrating the truth of the original statement \( \tan(\theta) \cot(\theta)=1\).

Step 1: Understand the basic identities

Recall the basic trigonometric identity which states that \(\tan(\theta) = \frac{1}{\cot(\theta)}\). This will be instrumental in transforming the given equation. Also, know that the multiplication of a number with its reciprocal will always equate to 1.

Step 2: Substitute cotangent identity

By substituting \(\cot(\theta)\) in terms of \(\tan(\theta)\) in the equation, we transform \(\tan(\theta) \cot(\theta)=1\) to \(\tan(\theta) * \frac{1}{\tan(\theta)} = 1\)

Step 3: Simplify the equation

Now simplify the transformed equation \(\tan(\theta) * \frac{1}{\tan(\theta)}\) which results in \(1\). Hence the right side equals to the left side of the original equation. Thus, the given equation \( \tan(\theta) \cot(\theta)=1 \) has been proven.

Key Concepts

Tangent Function

The tangent function is a fundamental concept in trigonometry. It's represented by \( \tan(\theta) \), which is the ratio of the opposite side to the adjacent side in a right triangle. This ratio makes tangent useful in numerous applications, like solving triangles and understanding wave behaviors. The tangent function has several interesting properties, such as being periodic with a period of \( \pi \).
In the context of the exercise, the property \( \tan(\theta) = \frac{1}{\cot(\theta)} \) is particularly important. This stems from the idea that tangent and cotangent are reciprocal functions of each other. When \( \tan(\theta) \) is multiplied by \( \cot(\theta) \), their product is always 1. This reciprocal property simplifies equations, especially when proving identities or solving problems.

Cotangent Function

The cotangent function, denoted as \( \cot(\theta) \), is closely related to the tangent function. It is defined as the reciprocal of the tangent function. This can be expressed as \( \cot(\theta) = \frac{1}{\tan(\theta)} \). In a right triangle, \( \cot(\theta) \) can also be seen as the ratio of the adjacent side to the opposite side.
Understanding the cotangent is crucial when dealing with trigonometric identities. It often allows for transforming and simplifying expressions, just like in our equation \( \tan(\theta) \cot(\theta) = 1 \). The property that cotangent is the reciprocal of tangent is key to the simplification steps described in the exercise.

Trigonometric Equation

Trigonometric equations are equations involving trigonometric functions, like sine, cosine, tangent, and their reciprocals. Solving these equations often involves applying trigonometric identities to simplify and transform one side of the equation to match the other, as seen in our exercise.
Some common strategies include:

• Using identities to rewrite the terms to reveal a pattern or simplify a complex equation.
• Recognizing reciprocal relationships, such as those seen between tangent and cotangent.
• Ensuring solutions are within the given domain; for our problem, this is \( 0 < \theta < \frac{\pi}{2} \).

These strategies are essential because trigonometric equations appear everywhere in math and physics, representing oscillations, waves, and rotational movements. Solving such equations is a valuable skill in the toolkit of anyone working with theoretical or applied mathematics.