Question

Prove each identity. $$\cos x=\sin \left(x+90^{\circ}\right)$$

Short answer

\( \cos x = \sin (x + 90^\circ) \) is true based on the co-function identities for complementary angles.

Step-by-step solution

Understand the Co-function identities

Co-function identities relate the sine and cosine of complementary angles. Remember that two angles are complementary if their measures add up to 90 degrees. The co-function identity for sine and cosine is: \( \text{sin}(90^\circ - \theta) = \text{cos}(\theta) \).

Apply the Co-function identity to the given expression

Apply the co-function identity to \( \text{sin}(x + 90^\circ) \) using \( \theta = x \). According to the identity, \( \text{sin}(90^\circ - \theta) = \text{cos}(\theta) \text{.} \) Here, replace \( \theta \) with \( x \) to get \( \text{sin}(90^\circ - x) \). However, to match the expression we are given, we need to consider \( \text{sin}(x + 90^\circ) \text{,} \) which is equivalent to \( \text{sin}(90^\circ + x) \text{.} \) By the periodic properties of sine, we know that \( \text{sin}(90^\circ + x) = \text{cos}(x) \).

Conclude the proof

Having shown that \( \text{sin}(x + 90^\circ) \) is equivalent to \( \text{cos}(x) \) using the co-function identities, we have thus proved the given identity.

Key concepts

Co-function Identities

Co-function identities are a cornerstone in understanding the relationship between trigonometric functions, particularly between sine and cosine. These identities showcase a unique symmetry: the sine of an angle is equal to the cosine of its complement, and vice versa. A complementary angle pairs add up to a right angle, or in terms of measure, 90 degrees.

Written mathematically, the co-function identities appear as: \< \cos(90^\circ - \theta) = \sin(\theta) \> and \< \sin(90^\circ - \theta) = \cos(\theta) \>. They are inversely related – when you subtract an angle from 90 degrees and then take the cosine, you get the same value as taking the sine of the original angle, and this holds true for all angle measures, provided both angles are in the same trigonometric quadrant.

These identities are pivotal for solving many trigonometric equations and proofs, as they can often simplify or transform expressions in a beneficial way. For instance, they reveal that for any angle \(\theta\), the sine of \(\theta\) is equal to the cosine of the angle that completes \(\theta\) to 90 degrees, thus bridging the gap between these two important trigonometric functions.

Understanding Complements

Whenever two angles are described as complementary, they always add up to a right angle. The importance of co-function identities is particularly apparent when working with complementary angles, as they allow the interchange between sine and cosine functions, streamlining the process of trigonometric calculations.

Complementary Angles

Complementary angles play an essential role in the realm of trigonometry, impacting the functions and identities significantly. When two angles are said to be complementary, they cumulatively measure up to 90 degrees. This characteristic is pivotal in understanding and applying the co-function identities.

For any angle \( \alpha \), its complement is given by \( 90^\circ - \alpha \). This relationship is not only geometric in nature but also influences trigonometric functions. For example, if we have an angle \( \alpha \) and want to find the sine of its complementary angle, we can simply use the cosine of \( \alpha \), due to the co-function identities.

Complementary angles teach us that trigonometric values are not isolated; rather, they are inherently linked through these specific relationships. The concept is not limited to just sine and cosine but extends to other pairs of trigonometric functions as well, contributing to a comprehensive and interconnected system that describes various properties of angles and triangles.

Periodic Properties of Sine

The periodic properties of the sine function underscore one of the key characteristics of trigonometry—its cyclic nature. Sine, being a periodic function, repeats its values in a regular interval known as the period. For the sine function, this period is \(360^\circ\) or \(2\pi\) radians, meaning that after every complete cycle, the sine function will return to the same value.

The significance of this periodicity is most apparent when working with angles that exceed a full circle, where the concept of angles as rotations becomes handy. Using periodic properties, one can determine the sine of an angle beyond \(360^\circ\) by subtracting complete cycles until an equivalent angle that is less than one full circle is found. Here's an expression that captures this principle: \< \sin(\theta \pm n\times360^\circ) = \sin(\theta) \>, where \(n\) is an integer.

Periodicity simplifies our work with trigonometric functions because it allows us to convert large angles to their acute representatives. As in our original exercise, if we encounter an expression like \(\sin(x + 90^\circ)\), we can confidently say that it equals the cosine of \(x\), as the sine function, after completing a quarter cycle (90 degrees), becomes the cosine function for a given angle. This property reveals that the sine function is intimately tied to the concept of cycles, rotations, and, as we've seen earlier, to the geometry of complementary angles.