Problem 41 - Calculus Single Variable

Question

Use one or more of the basic trigonometric identities to derive the given identity. \(\sin (\theta)=\cos \left(\frac{\pi}{2}-\theta\right)\)

Step-by-Step Solution

Answer

The identity \(\sin(\theta) = \cos(\frac{\pi}{2} - \theta)\) is derived from the complementary angle relationship.

Step 1: Review Basic Trigonometric Identities

Recall that one of the basic trigonometric identities involves using complementary angles: \(\sin(\theta) = \cos(\frac{\pi}{2} - \theta)\). This identity states that the sine of an angle is equal to the cosine of its complementary angle.

Step 2: Define the Complementary Angle

A complementary angle is one where the sum of the angles adds up to \(\frac{\pi}{2}\). In this case, for an angle \(\theta\), the complementary angle is \(\frac{\pi}{2} - \theta\).

Step 3: Substitute and Verify

If \(\alpha = \theta\), then by the identity \(\sin(\alpha) = \cos(\frac{\pi}{2} - \alpha)\). Substitute \(\theta\) for \(\alpha\), so you get \(\sin(\theta) = \cos(\frac{\pi}{2} - \theta)\), which is the identity we started with. This confirms that \(\sin(\theta) = \cos(\frac{\pi}{2} - \theta)\).

Key Concepts

Complementary Angles

In trigonometry, complementary angles are two angles whose measures add up to \(\frac{\pi}{2}\) radians, which is equivalent to 90 degrees. This means that if you have an angle denoted as \(\theta\), its complementary angle can be easily found by the formula \(\frac{\pi}{2} - \theta\). Complementary angles hold a special place in trigonometry as they are foundational to understanding various trigonometric identities, particularly the relationships between sine and cosine functions.

• For example, if \(\theta = \frac{\pi}{6}\) (which is 30 degrees), its complementary angle would be \(\frac{\pi}{2} - \frac{\pi}{6} = \frac{\pi}{3}\) (which is 60 degrees).
• In a right triangle, the two non-right angles are always complementary.

Understanding complementary angles is crucial, especially when dealing with problems that involve the Pythagorean identity or other derived identities. It serves as an essential building block for more complex topics in trigonometry.

Sine and Cosine Relationship

One of the beautiful aspects of trigonometry is the inherent relationship between the sine and cosine functions. This relationship is succinctly captured in the identity \(\sin(\theta) = \cos(\frac{\pi}{2} - \theta)\). This formula emerges directly from the definition of complementary angles and is instrumental in many trigonometric derivations and simplifications.

• The sine of an angle is equal to the cosine of its complementary angle.
• This relationship can be visualized on the unit circle, where the \(x\)-coordinate (cosine) of one angle corresponds to the \(y\)-coordinate (sine) of its complementary angle.
• This property can be used to transform one trigonometric function into another, which is particularly useful in integration and differentiation in calculus.

By applying this concept, you can not only verify the authenticity of trigonometric identities but also simplify expressions involving trigonometric functions.

Basic Trigonometry

Basic trigonometry forms the cornerstone of understanding and working with angles and sides in various mathematical contexts. At its core, trigonometry involves the relationships between the sides and angles of triangles. The three primary trigonometric functions are sine, cosine, and tangent, often abbreviated as \(\sin\), \(\cos\), and \(\tan\).

• Each function has a unique role: \(\sin(\theta)\) is the ratio of the opposite side to the hypotenuse in a right triangle, \(\cos(\theta)\) is the ratio of the adjacent side to the hypotenuse, and \(\tan(\theta)\) is the ratio of the opposite side to the adjacent side.
• These functions can also be represented as coordinates on the unit circle, which provides a geometric interpretation that extends beyond the right triangle.
• Trigonometry is not just limited to theoretical mathematics; it has practical applications in physics, engineering, and even in fields like music and art.

Developing a solid understanding of basic trigonometry is pivotal, as it paves the way for exploring more advanced mathematical theories and principles.