Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-22 :\) a. Rewrite each function value in terms of its cofunction. b. Find, to four decimal places, the value of the function value found in a. $$ \tan 54^{\circ} $$

Step-by-Step Solution

Answer

\(\tan 54^{\circ} = \cot 36^{\circ} \approx 1.3764\)

Step 1: Understand Cofunction Identities

Cofunction identities exploit the complement rule in right triangles. For tangent and cotangent, we use: \[\tan(\theta) = \cot(90^\circ - \theta)\] This identity will allow us to express the tangent function in terms of its cofunction, the cotangent.

Step 2: Apply the Cofunction Identity

Given \(\tan 54^{\circ}\), use the identity for tangent: \[\tan(54^{\circ}) = \cot(90^\circ - 54^{\circ})\]Calculate the angle: \[90^\circ - 54^\circ = 36^\circ\]Thus, \(\tan(54^{\circ}) = \cot(36^{\circ})\).

Step 3: Calculate the Cofunction Value

Find the cotangent of 36 degrees using a calculator:\[\cot(36^{\circ}) = \frac{1}{\tan(36^{\circ})}\]Calculate \(\tan(36^{\circ})\) using a calculator, which is approximately \(0.7265\). Then, compute the reciprocal:\[\cot(36^{\circ}) = \frac{1}{0.7265} \approx 1.3764\]

Step 4: Present the Result

Therefore, \(\tan(54^{\circ}) \) expressed as its cofunction \(\cot(36^{\circ}) \) is approximately \(1.3764\).

Key Concepts

Tangent

The tangent of an angle in a right triangle is a fundamental trigonometric function. It relates the lengths of the sides of the triangle to the angle. Specifically, the tangent function is defined as the ratio of the opposite side to the adjacent side of a given angle:

• The formula is: \( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \)
• The angle \( \theta \) is one of the non-right angles in a right triangle.

For example, consider a right triangle where the side opposite the angle \(\theta\) is 5 units long, and the side adjacent to \(\theta\) is 3 units long. The tangent of \(\theta\) would be \(\frac{5}{3}\).
The tangent function is periodic, repeating every 180 degrees or \(\pi\) radians. This means \(\tan(\theta) = \tan(\theta + n \times 180^{\circ})\), where \(n\) is an integer. The tangent function is also undefined at odd multiples of 90 degrees due to the division by zero (the adjacent side equals zero).
In trigonometry, the tangent function often pairs with its reciprocal function, cotangent, which brings us to the next topic.

Cotangent

Cotangent is closely related to the tangent function. As the reciprocal of tangent, cotangent is another trigonometric function that connects the sides of a right triangle to its angles. It is expressed as:

• \( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{Adjacent}}{\text{Opposite}} \)
• Like tangent, \( \theta \) is an angle in a right triangle.

Cotangent is excellent for situations where you need to switch perspectives from opposite to adjacent roles of sides. Imagine a triangle where the length of the adjacent side to an angle \( \theta \) is 3 units, and the opposite side is 5 units; here, \( \cot(\theta) = \frac{3}{5}\).
Cotangent helps simplify problems that use or focus on tangent-heavy expressions. It is also periodic, with the period being 180 degrees or \( \pi \) radians, much like the tangent function.
Certain angles, especially multiples of 90 degrees, where tangent is undefined, will provide specific boundaries and values for cotangent as well, emphasizing their interconnected nature.

Right Triangle Cofunction

Cofunction identities are crucial in trigonometry for understanding the interrelatedness of different trigonometric functions. For right angles, these identities encapsulate how the trigonometric function of one angle corresponds to that of its complement. For tangent and cotangent, the identity is:

• \( \tan(\theta) = \cot(90^{\circ} - \theta) \)
• \( \cot(\theta) = \tan(90^{\circ} - \theta) \)

In the context of a right triangle, the two non-right angles of the triangle sum up to 90 degrees. This means if one angle is \(\theta\), the other non-right angle is \(90^{\circ} - \theta\).
This relationship is what makes cofunction identities so powerful, as they allow you to seamlessly switch between trigonometric functions based solely on angle complements.
For a practical example, if \( \tan(54^{\circ})\) is given, as seen in our exercise, using the cofunction identity directly leads us to \( \cot(36^{\circ}) \), providing an effective computational shortcut that simplifies problems involving right angle trigonometry.