Question

Cofunctions. Express as a function of the complementary angle. $$\sin 38^{\circ}$$

Short answer

\(\sin(38^\circ) = \cos(52^\circ)\)

Step-by-step solution

Understanding Cofunctions

Cofunctions are pairs of trigonometric functions that are equal when their angles sum to 90 degrees. The sine and cosine are cofunctions. Therefore, the sine of an angle is equal to the cosine of its complementary angle.

Finding the Complementary Angle

To find the complementary angle, subtract the given angle from 90 degrees. For the given angle, the complementary angle is \(90^\circ - 38^\circ = 52^\circ\).

Expressing Sin as the Cosine of the Complementary Angle

Using the definition of cofunctions, we can express \(\sin(38^\circ)\) as \(\cos(90^\circ - 38^\circ)\), which is \(\cos(52^\circ)\). Therefore, \(\sin(38^\circ)\) is equal to \(\cos(52^\circ)\).

Key concepts

Complementary Angles

Complementary angles play a crucial role in our understanding of trigonometry. They are defined as two angles whose sum equals 90 degrees, effectively forming a right angle when combined. This concept is not only useful in geometry but also serves as the foundation for understanding cofunctions in trigonometry.

For instance, if we have an angle measuring 38 degrees, its complementary angle will be obtained by deducting it from 90 degrees. Mathematically, we represent it as: \[90 degrees - 38 degrees = 52 degrees\]. The angles 38 degrees and 52 degrees are complementary because they add up to a right angle. Recognizing complementary angles helps in solving various trigonometric problems and is particularly essential when working with sine and cosine functions, as they are cofunctions of each other when dealing with complementary angles.

Sine and Cosine Relationship

In trigonometry, the relationship between sine and cosine is fascinating as it shows how interconnected these functions are. The sine of an angle is the cosine of the complementary angle, and vice versa. This is because they both represent ratios of sides of a right triangle, but from different perspectives. When one function increases, the other decreases, reflecting their dependency on the angle's size.

To understand this with an example, consider a right triangle with an angle of 38 degrees. The sine of this angle, \(\sin(38^\circ)\), can be written as the cosine of its complementary angle, \(\cos(52^\circ)\), because \[52^\circ\] is the complement of \[38^\circ\]. This sine and cosine relationship is expressed mathematically by the cofunction identity: \[\sin(\theta) = \cos(90^\circ - \theta)\]. It is a powerful tool that simplifies many trigonometric expressions and solutions.

Trigonometric Functions

Trigonometric functions are the backbone of trigonometry, as they relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan), among others. Each function provides a unique ratio based on a right-angled triangle: sine represents the ratio of the opposite side to the hypotenuse, cosine that of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side.

The functions extend beyond right-angled triangles and are defined for all angles, offering a way to describe periodic phenomena through their graphs. Additionally, trigonometric functions are vital in various fields including physics, engineering, and even music theory. Learning how these functions work and relate to each other not only helps in solving mathematical problems but also in understanding the cyclical nature of many real-world phenomena.