Prove the following for all x ∈ℝ. sin(π-x)=sinx, sin((π/ 2)-x)=cosx, sin((π/ 2)+x)=cosx, cos

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Prove the following for all x ∈ℝ. sin(π-x)=sinx, sin((π/...

Prove the following for all $x \in \mathbb{R}$. $\sin (\pi-x)=\sin x, \quad \sin ((\pi / 2)-x)=\cos x, \quad \sin ((\pi / 2)+x)=\cos x$,
$\cos (\pi-x)=-\cos x, \quad \cos ((\pi / 2)-x)=\sin x, \quad \cos ((\pi / 2)+x)=-\sin x .$

Key Concepts

Unit Circle Interpretation

The unit circle is central to understanding trigonometric functions. It provides a geometric representation where angles correspond to points on the circle, and the sine and cosine of these angles represent the y and x coordinates, respectively. This interpretation facilitates the visualization of how transformations such as shifts by ? or ?/2 affect the values of sine and cosine by reflecting points across axes or rotating them around the circle.

Sum and Difference Formulas

The sum and difference formulas for sine and cosine are pivotal in proving many trigonometric identities. They allow the expression of the sine or cosine of a sum or difference of angles in terms of the sines and cosines of the individual angles. These formulas are used to decompose angles like ? ? x or (?/2) ± x, setting the groundwork for proving equalities between different trigonometric expressions.

Co-Function Identities

Co-function identities show the relationship between pairs of trigonometric functions when the angles are complementary. For example, sin(?/2 ? x) = cos x and cos(?/2 ? x) = sin x highlight the intrinsic symmetry between sine and cosine. This concept is crucial in understanding how the functions transform under angle shifts, simplifying various trigonometric expressions and proofs.

Symmetry and Reflection Properties

Symmetry and reflection properties of trigonometric functions involve understanding how these functions behave under reflections and rotations. For instance, the identity sin(? ? x) = sin x demonstrates the reflection symmetry of the sine function about the line x = ?/2, while cos(? ? x) = -cos x shows how cosine changes sign under similar transformations. Recognizing these symmetry properties provides a powerful tool for proving trigonometric identities without resorting solely to algebraic manipulation.

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