Use the Relationship Between Sine and Cosine of Complementary Angles

Introduction

Key Concepts

Complementary Angles in Trigonometry

In trigonometry, two angles are said to be complementary if the sum of their measures is 90 degrees. If one angle is denoted by $\theta$, its complementary angle is $(90^\circ - \theta)$. This relationship is crucial because it allows us to express sine and cosine functions of complementary angles in terms of each other.

Sine and Cosine Functions

The sine and cosine functions are fundamental trigonometric functions defined for an angle $\theta$ in a right-angled triangle. For an angle $\theta$:

• Sine: $\sin(\theta) = \frac{\text{Opposite Side}}{\text{Hypotenuse}}$
• Cosine: $\cos(\theta) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}$

Relationship Between Sine and Cosine of Complementary Angles

The key relationship is expressed by the co-function identities:

• $\sin(90^\circ - \theta) = \cos(\theta)$
• $\cos(90^\circ - \theta) = \sin(\theta)$

These identities show that the sine of an angle is equal to the cosine of its complementary angle, and vice versa. This is derived from the complementary nature of the angles in a right-angled triangle.

Derivation of Co-function Identities

Consider a right-angled triangle where one acute angle is $\theta$, and the other acute angle is $90^\circ - \theta$. Let the sides opposite to these angles be denoted as $a$ and $b$, respectively, and the hypotenuse as $h$.

By definition:

• $\sin(\theta) = \frac{a}{h}$
• $\cos(\theta) = \frac{b}{h}$
• $\sin(90^\circ - \theta) = \frac{b}{h}$
• $\cos(90^\circ - \theta) = \frac{a}{h}$

From the above, it follows that:

• $\sin(\theta) = \cos(90^\circ - \theta)$
• $\cos(\theta) = \sin(90^\circ - \theta)$

Graphical Interpretation

On the unit circle, an angle $\theta$ and its complement $90^\circ - \theta$ correspond to coordinates that reflect the relationship between sine and cosine. Specifically, the sine of an angle corresponds to the cosine of its complement, illustrating the co-function identities graphically.

Examples and Applications

Consider an angle $\theta = 30^\circ$. Its complementary angle is $60^\circ$. Applying the co-function identities:

• $\sin(30^\circ) = \frac{1}{2}$
• $\cos(60^\circ) = \frac{1}{2}$

Thus, $\sin(30^\circ) = \cos(60^\circ)$, reaffirming the relationship. This property is useful in simplifying trigonometric expressions and solving equations involving complementary angles.

Mathematical Proof Using Unit Circle

On the unit circle, any angle $\theta$ has coordinates $(\cos(\theta), \sin(\theta))$. The complementary angle $90^\circ - \theta$ has coordinates $(\cos(90^\circ - \theta), \sin(90^\circ - \theta))$. Given the co-function identities:

• $\sin(90^\circ - \theta) = \cos(\theta)$
• $\cos(90^\circ - \theta) = \sin(\theta)$

This symmetry in the unit circle further validates the relationship between sine and cosine of complementary angles.

Using Co-function Identities in Equations

Co-function identities are instrumental in solving trigonometric equations. For instance, to solve $\sin(\theta) = \cos(45^\circ)$ for $\theta$, we can use the identity:

Setting $\cos(90^\circ - \theta) = \cos(45^\circ)$ leads to $90^\circ - \theta = 45^\circ$, hence $\theta = 45^\circ$.

Inverse Relationships

The inverse relationships can also be expressed as:

• $\theta = 90^\circ - \sin^{-1}(\cos(\theta))$
• $\theta = 90^\circ - \cos^{-1}(\sin(\theta))$

These are useful in determining angles when one of the trigonometric functions is known.

Applications in Real-world Problems

Understanding complementary angles and their trigonometric relationships is essential in fields such as engineering, physics, and architecture. For example, determining forces in equilibrium often involves resolving components using sine and cosine functions of complementary angles.

Summary of Key Concepts

• Definition of complementary angles: Two angles whose measures add up to $90^\circ$.
• Co-function identities: $\sin(90^\circ - \theta) = \cos(\theta)$ and $\cos(90^\circ - \theta) = \sin(\theta)$.
• Derivation using right-angled triangles and the unit circle.
• Applications in simplifying trigonometric expressions and solving equations.

Advanced Concepts

Extension to Radians

While the primary discussion is in degrees, the relationship extends to radian measures. Since $90^\circ$ is equivalent to $\frac{\pi}{2}$ radians, the co-function identities become:

• $\sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta)$
• $\cos\left(\frac{\pi}{2} - \theta\right) = \sin(\theta)$

This extension is crucial for applications in higher mathematics and engineering where radian measures are standard.

General Co-function Identities

The concept of co-function identities is not limited to sine and cosine. Other trigonometric functions also have co-function relationships:

• $\tan(90^\circ - \theta) = \cot(\theta)$
• $\cot(90^\circ - \theta) = \tan(\theta)$
• $\sec(90^\circ - \theta) = \csc(\theta)$
• $\csc(90^\circ - \theta) = \sec(\theta)$

These identities are derived similarly and are essential for solving more complex trigonometric problems.

Applications in Calculus

In calculus, understanding the relationship between sine and cosine of complementary angles is vital for differentiation and integration involving trigonometric functions. For example, the derivative of $\sin(\theta)$ is $\cos(\theta)$, which aligns with the co-function identity when considering complementary angles.

Solving Trigonometric Equations

Advanced problem-solving often involves equations such as:

Using co-function identities, we can transform and solve these equations by expressing all terms in terms of a single trigonometric function.

Trigonometric Identities and Transformations

The co-function identities are part of a broader set of trigonometric identities that facilitate the simplification and transformation of expressions. They are foundational for deriving other important identities like the Pythagorean identities and angle addition formulas.

Interdisciplinary Connections

The relationship between sine and cosine of complementary angles bridges various disciplines. In physics, it plays a role in analyzing waveforms and oscillations. In engineering, it's essential for designing structures and electrical circuits. Additionally, in computer graphics, these trigonometric relationships are used to model rotations and transformations.

Complex Problem Example

Problem: Given that $\sin(\theta) = \frac{3}{5}$, find $\cos(90^\circ - \theta)$ and verify using the co-function identity.

Solution:

• Given $\sin(\theta) = \frac{3}{5}$
• Using the co-function identity: $\cos(90^\circ - \theta) = \sin(\theta)$
• Therefore, $\cos(90^\circ - \theta) = \frac{3}{5}$
• Verification: Calculate $\cos(90^\circ - \theta)$ directly.

Let's assume $\theta = 36.87^\circ$, since $\sin(36.87^\circ) \approx \frac{3}{5}$. Then:

• $\cos(90^\circ - 36.87^\circ) = \cos(53.13^\circ) \approx \frac{3}{5}$

This confirms the co-function identity.

Inverse Trigonometric Functions and Complementary Angles

Inverse trigonometric functions, such as $\sin^{-1}(x)$ and $\cos^{-1}(x)$, also exhibit relationships based on complementary angles. For example:

This identity demonstrates that the angles whose sine and cosine are $x$ add up to 90 degrees (or $\frac{\pi}{2}$ radians), reflecting the complementary nature of their functions.

Co-function Identities in Vector Analysis

In vector analysis, co-function identities assist in resolving vector components. When decomposing vectors into perpendicular components, the angles involved often sum to 90 degrees, allowing the use of these identities for simplified calculations.

Applications in Electrical Engineering

In electrical engineering, alternating current (AC) circuits involve phase angles where voltage and current may be out of phase by complementary angles. Understanding the sine and cosine relationships of these angles is essential for analyzing power factors and impedance in circuits.

Integration Techniques Using Co-function Identities

When integrating products of sine and cosine functions, co-function identities can simplify the integrand, making the integration process more straightforward. For example:

Using the identity $\sin(\theta) = \cos(90^\circ - \theta)$, we can apply substitution methods to solve the integral.

Higher Order Identities

Co-function identities extend to higher-order trigonometric functions and can be combined with multiple-angle formulas to derive more complex identities. These are useful in advanced mathematical proofs and problem-solving scenarios.

Summary of Advanced Concepts

• Extension of co-function identities to radians.
• General co-function identities for all trigonometric functions.
• Applications in calculus, physics, engineering, and computer graphics.
• Advanced problem-solving techniques utilizing co-function identities.
• Interdisciplinary connections enhancing the applicability of trigonometric relationships.

Comparison Table

Summary and Key Takeaways