Problem 37 - Algebra and Trigonometry

Question

Find a cofunction with the same value as the given expression. $$ \cos \frac{2 \pi}{5} $$

Step-by-Step Solution

Answer

The cofunction with the same value as the given expression \( \cos \frac{2 \pi}{5} \) is \( \sin \frac{2 \pi}{5} \).

Step 1: Understanding the cofunction identity

Cosine and sine are cofunctions, meaning their values for complementary angles are equal. This can be written as \( \sin(\frac{\pi}{2} - x) = \cos(x) \). Therefore, in order to find an equivalent sine expression for \( \cos \frac{2 \pi}{5} \), we need to express \( \frac{2 \pi}{5} \) as \( \frac{\pi}{2} - x \).

Step 2: Finding the Complementary Angle

Firstly, we know that a quarter of a full rotation in radians is \( \frac{\pi}{2} \). We want to find a value of \( x \) such that \( \frac{\pi}{2} - x = \frac{2 \pi}{5} \). Solving this for \( x \) gives \( x = \frac{\pi}{2} - \frac{2 \pi}{5} = \frac{5 \pi - 4 \pi}{10} = \frac{\pi}{10} \).

Step 3: Replacing with the Cofunction

Now by substituting \( x = \frac{\pi}{10} \) into the identity \( \sin(\frac{\pi}{2} - x) = \cos(x) \), we get the equivalent sine expression: \( \sin(\frac{\pi}{2} - \frac{\pi}{10}) = \sin(\frac{4 \pi}{10}) = \sin(\frac{2\pi}{5}) \). Thus, \( \sin(\frac{2\pi}{5}) \) is a cofunction with the same value as \( \cos(\frac{2\pi}{5}) \).

Key Concepts

Complementary Angles

Complementary angles are two angles whose measures add up to precisely 90 degrees or, in radian measure, \( \frac{\pi}{2} \) radians. Understanding this concept is vital in trigonometry because certain trigonometric identities rely on the relationship between these angles. When two angles are complementary, the trigonometric functions of one angle can often be expressed in terms of the cofunctions of the other angle.Here are a few essential points about complementary angles:

• If angle \( A \) is \( \frac{\pi}{4} \), then its complement, angle \( B \), would be \( \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4} \).
• The two angles combined always make a right angle.
• They appear often in geometry and trigonometry problems involving cofunction identities.

This special relationship allows us to solve and simplify expressions using trigonometric functions like sine and cosine effectively.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, specifically in dealing with angles and side lengths in triangles. The primary trigonometric functions include sine (sin), cosine (cos), and tangent (tan). Each of these functions relates to the ratios of the sides of right-angled triangles.To understand their importance:

• The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse.
• The cosine of an angle is the ratio of the adjacent side to the hypotenuse.
• The tangent is the ratio of the opposite side to the adjacent side.

Trigonometric functions are periodically repeating, meaning they cycle through their values at regular intervals. This repetition makes them invaluable in modeling periodic phenomena like sound waves or circular motion.The cofunction identity for cosine and sine states that the sine of an angle is equal to the cosine of its complement, or \( \sin(\frac{\pi}{2} - x) = \cos(x) \). This identity is particularly useful for solving equations where one wants to express a cosine term in terms of a sine function, or vice versa.

Radian Measure

Radian measure is the standard unit of angular measure used in many branches of mathematics. Unlike degrees, which divide a circle into 360 parts, radians divide the circle according to the arc length, making them more naturally suited for mathematical calculations involving circles.Key aspects of radian measure include:

• A full circle is \( 2\pi \) radians.
• A half-circle, or a straight angle, measures \( \pi \) radians.
• A right angle, or a quarter of a circle, measures \( \frac{\pi}{2} \) radians.

Using radian measure simplifies many mathematical expressions. For example, when converting between degrees and radians, one uses the formula \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \) or \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). This conversion is essential because most trigonometric functions and identities are conventionally expressed using radians.Radian measure is frequently used when dealing with calculus and trigonometry problems, offering a seamless connection between angles and the arc lengths they subtend.