Problem 51 - Precalculus : Building Concepts and Connections

Question

Find a cofunction that has the same value as the given quantity. $$\cot 67^{\circ}$$

Step-by-Step Solution

Answer

The cofunction that has the same value as \( \cot 67^{\circ} \) is \( \cot 23^{\circ} \).

Step 1: Understanding the concept of cofunctions

A cofunction in trigonometry is a function of an angle that is the complement of another function. For an angle \( \alpha \), the cotangent of \( \alpha \) is equal to the cotangent of \( 90 – \alpha \) degrees. We use this relationship to find the cofunction that has the same value as \( \cot 67^{\circ} \).

Step 2: Applying the relationship

Subtract the given angle from 90 degrees as the cotangent and its cofunction add up to 90 degrees. Therefore, the cofunction becomes \( \cot (90 - 67) = \cot 23^{\circ} \).

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are fundamental to trigonometry, a branch of mathematics that studies the relationships between angles and sides in triangles. There are six primary trigonometric functions: sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)).

• Sine: Represents the ratio of the length of the opposite side to the length of the hypotenuse.
• Cosine: Represents the ratio of the length of the adjacent side to the length of the hypotenuse.
• Tangent: Represents the ratio of the length of the opposite side to the length of the adjacent side.
• Cotangent: The reciprocal of tangent, meaning it represents the ratio of the adjacent side to the opposite side.
• Cosecant: The reciprocal of sine.
• Secant: The reciprocal of cosine.

These functions are essential in various fields, including engineering, physics, and architecture, as they help solve problems related to angles and dimensions. They can be visualized on the unit circle, making it easier to understand their behavior for different angles.

Cofunctions in Trigonometry

Cofunctions are paired trigonometric functions that relate the angles summing to 90 degrees in right triangles. Each function has a cofunction, which is another trigonometric function.

• Sine and Cosine: The cofunction of sine (\(\sin\)) is cosine (\(\cos\)): \( \sin(90^\circ - \theta) = \cos(\theta) \)
• Tangent and Cotangent: The cofunction of tangent (\(\tan\)) is cotangent (\(\cot\)): \( \tan(90^\circ - \theta) = \cot(\theta) \)
• Secant and Cosecant: The cofunction of secant (\(\sec\)) is cosecant (\(\csc\)): \( \sec(90^\circ - \theta) = \csc(\theta) \)

Cofunction identities are useful for transforming and simplifying expressions when dealing with complementary angles. In the case of \(\cot 67^\circ\), its cofunction \(\cot 23^\circ\) has the same value because \(67^\circ + 23^\circ = 90^\circ\). This relationship is specifically handy in solving trigonometric equations and verifying identities.

Understanding Angle Measurement

Angle measurement is a crucial concept in trigonometry. Angles are typically measured in degrees or radians, each with its own advantages.Degrees are a more intuitive way to measure angles, especially for beginners. One full rotation equals 360 degrees. Most day-to-day activities, like setting angles in geometry or measuring circles, use degrees due to their simplicity.Radians, on the other hand, offer a mathematical advantage, especially in calculus. One full rotation equals \(2\pi\) radians. They are defined based on the radius of a circle, making them a remarkable tool for mathematical calculations that involve periodic functions.Ultimately, choosing between degrees and radians depends on the context. It's crucial to understand and comfortably switch between these units based on the specific requirements of the mathematical problem at hand. Converting between them is straightforward, as \(1^\circ\) corresponds to \(\frac{\pi}{180}\) radians and vice versa.