Problem 37 - A Graphical Approach to Precalculus with Limits

Question

Write each expression in terms of its co-function. $$\cot 73^{\circ}$$

Step-by-Step Solution

Answer

\( \cot 73^{\circ} \) is \( \tan 17^{\circ} \).

Step 1: Understand the Concept of Co-Functions

In trigonometry, co-functions relate the angles that add up to 90 degrees. The basic co-function identities include \( \sin(\theta) = \cos(90^{\circ} - \theta) \), \( \cos(\theta) = \sin(90^{\circ} - \theta) \), \( \tan(\theta) = \cot(90^{\circ} - \theta) \), and \( \cot(\theta) = \tan(90^{\circ} - \theta) \). We will use the co-function identity for cotangent for this problem.

Step 2: Identify the Appropriate Co-Function Identity

The problem requires expressing \( \cot 73^{\circ} \) in terms of its co-function. From the co-function identities, \( \cot(\theta) = \tan(90^{\circ} - \theta) \). Therefore, \( \cot 73^{\circ} \) can be expressed as \( \tan(90^{\circ} - 73^{\circ}) \).

Step 3: Calculate \( 90^{\circ} - 73^{\circ} \)

Find the value of \( 90^{\circ} - 73^{\circ} \). This is the complement of 73 degrees: \( 90^{\circ} - 73^{\circ} = 17^{\circ} \).

Step 4: Write \( \cot 73^{\circ} \) in Terms of its Co-Function

Substitute the calculated result from Step 3 into the co-function identity. Thus, \( \cot 73^{\circ} = \tan 17^{\circ} \).

Key Concepts

Trigonometry

Trigonometry is a fascinating branch of mathematics that deals with the relationships between the angles and sides of triangles. It's most commonly used with right-angled triangles, where one of the angles is 90 degrees. By leveraging trigonometric functions, we can determine known and unknown elements of these triangles.

Trigonometry is crucial in various fields such as physics, engineering, and architecture, as it allows us to calculate heights, distances, and angles accurately.

• Sine and cosine are the ratios of the sides of a right triangle relative to its angles.
• Tangent and cotangent are also important trigonometric functions, focusing on the slope formed by the angles.

Understanding these fundamental trigonometric concepts can boost problem-solving skills and open up a world of analytical possibilities.

Cotangent

The cotangent is one of the basic trigonometric functions and is denoted as \(\cot(\theta)\). It is the reciprocal of the tangent function. In terms of a right triangle, it is the ratio of the length of the adjacent side to the opposite side.

Mathematically, this can be expressed as:\[\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{adjacent}}{\text{opposite}}\]Cotangent is particularly useful in various trigonometric transformations and simplifications. One interesting feature of cotangent is how it relates to tangent through co-function identities.

• Cotangent and tangent are co-functions, meaning:
• \(\cot(\theta) = \tan(90^{\circ} - \theta)\)

This relationship helps in reducing complex trigonometric expressions and solving various trigonometric equations.

Tangent

The tangent function, often abbreviated as \(\tan\), is a fundamental trigonometric function representing the ratio of the length of the opposite side to the adjacent side of a right triangle. This can be mathematically expressed as: \[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]

When examining angles in trigonometric contexts, the tangent function gives us a slope-like interpretation. Tangent is critical in many practical applications, from determining angles in construction projects to calculating slopes in land surveying.

• Tangent is closely related to cotangent, its co-function, through the identity:
• \(\tan(\theta) = \cot(90^{\circ} - \theta)\)

This identity allows mathematicians and scientists to solve trigonometric problems by translating angles into complementary perspectives, thereby simplifying complex geometrical analyses.