Problem 39 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \left(\tan 45^{\circ}\right)\left(\cot 45^{\circ}\right) $$

Step-by-Step Solution

Answer

The exact value is 1.

Step 1: Identify the tangent of 45 degrees

The tangent of 45 degrees is a well-known trigonometric value. It is given by \( \tan 45^{\circ} = 1 \). This means that the value of the tangent function at 45 degrees is 1.

Step 2: Identify the cotangent of 45 degrees

Similarly, cotangent of 45 degrees is the reciprocal of the tangent value. Therefore, \( \cot 45^{\circ} = 1 \) because the cotangent of 45 degrees is also 1.

Step 3: Multiply the values

Now, multiply the tangent and cotangent of 45 degrees together: \( 1 \times 1 = 1 \). So, the product of \( \tan 45^{\circ} \) and \( \cot 45^{\circ} \) is 1.

Key Concepts

Understanding the Tangent Function

The tangent function is one of the fundamental trigonometric functions and it plays a key role in understanding angles in right triangles. It is represented as \( \tan \theta \), with \( \theta \) being the angle. In a right triangle,

• Tangent is the ratio of the length of the opposite side to the adjacent side.
• The formula for tangent can be written as \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \).

When dealing with angles such as 45 degrees, the tangent has some particular properties.

• For \( \theta = 45^\circ \), \( \tan 45^\circ = 1 \) because the opposite and adjacent sides are equal in length.

This is an important trigonometric value to remember as it recurs frequently in both mathematics and applied sciences.
Understanding these properties helps in simplifying trigonometric expressions and solving equations effectively.

Exploring the Cotangent Function

The cotangent function is directly related to the tangent function; it is actually its reciprocal. It is denoted by \( \cot \theta \), and can be defined as the reciprocal of the tangent of an angle. In mathematical terms:

• \( \cot \theta = \frac{1}{\tan \theta} \)
• This means \( \cot \theta \) is also equal to \( \frac{\text{adjacent}}{\text{opposite}} \).

For angles like 45 degrees:

• \( \cot 45^\circ = \frac{1}{\tan 45^\circ} = 1 \)
• This relationship underscores why calculating cotangent for such an angle is straightforward and results in unity.

Recognizing this reciprocal nature allows for straightforward manipulation in equations involving trigonometric identities.

Reciprocal Identities in Trigonometry

Reciprocal identities are a powerful set of relationships in trigonometry that define one trigonometric function in terms of another. These are especially useful in solving complex trigonometric equations by transforming them into more manageable forms. For instance:

• The reciprocal identity for tangent is \( \cot \theta = \frac{1}{\tan \theta} \).
• This helps simplify expressions involving products or quotients of tangent and cotangent.

In the given exercise involving the multiplication of \( \tan 45^\circ \) and \( \cot 45^\circ \):

• The reciprocal relationship confirms the multiplication yields \( 1 \times 1 = 1 \).
• This elucidates that the interaction between the tangent and cotangent of the same angle generally results in 1, a clear-cut result thanks to their reciprocal nature.

Applying reciprocal identities can be an efficient strategy in trigonometry, aiding in reducing the complexity of equations and enhancing problem solving efficiency.