Problem 9 - 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. $$ \cot 57^{\circ} $$

Step-by-Step Solution

Answer

\( \cot 57^\circ = \tan 33^\circ \approx 0.6494 \)

Step 1: Understanding Cofunctions

The cofunction identities relate trigonometric functions. For example, \( \cot(\theta) = \tan(90^\circ - \theta) \). Our task is to rewrite \( \cot 57^\circ \) using this identity.

Step 2: Apply the Cofunction Identity

Using the identity for cotangent, we have: \( \cot(\theta) = \tan(90^\circ - \theta) \). Therefore, \( \cot 57^\circ = \tan(90^\circ - 57^\circ) \).

Step 3: Simplify the Angle

Calculate \( 90^\circ - 57^\circ = 33^\circ \). So \( \cot 57^\circ = \tan 33^\circ \).

Step 4: Find the Decimal Value

Use a calculator to determine \( \tan 33^\circ \) to four decimal places. \( \tan 33^\circ \approx 0.6494 \).

Step 5: Final Verification

Verify using a calculator that the decimal approximation is correct for four decimal places.

Key Concepts

Cotangent

The cotangent function is one of the foundational functions in trigonometry. It is the reciprocal of the tangent function. If we have a right triangle, the cotangent of an angle is defined as the length of the adjacent side divided by the length of the opposite side. Mathematically, it is expressed as:
\[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{adjacent}}{\text{opposite}} \]
The cotangent is often useful in trigonometry because it links the angle with its complementary angle through cofunction identities. Knowing about cotangent can help in solving various trigonometric equations and converting between functions, which is particularly handy when reducing complex expressions.

Tangent Function

The tangent function is another core component of trigonometry and plays a critical role in understanding angles and triangles. In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. It is given by:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
The tangent function measures how steep a line is and has periodic properties that are handy in modeling cyclical behaviors.

• The tangent function has a period of \(180^\circ\) or \(\pi\) radians.
• Its value ranges from \(-\infty\) to \(+\infty\).
• Understanding tangent is essential for solving relationships in trigonometry and calculus.

Because the tangent function is closely related to the cotangent, understanding one enhances your grasp of the other through reciprocal relationships.

Cofunction Identities

Cofunction identities are a set of trigonometric relationships that express a function in terms of its cofunction. These identities take advantage of complementary angles, which add up to \(90^\circ\) or \(\pi/2\) radians. The basic idea is:
\[ \cot(\theta) = \tan(90^\circ - \theta) \]
This identity simplifies converting between cotangent and tangent, as seen in the exercise provided where:

• \(\cot 57^\circ\) is transformed into \(\tan 33^\circ\).
• This rearrangement helps in utilizing relationships that might be more straightforward to compute or understand.

Cofunction identities are not limited to cotangent and tangent; they apply to all trigonometric functions. Embracing these identities allows for elegant solutions to complex trigonometric problems and offers deeper insight into the geometry of angles.