Origami: The Japanese art of origami involves the repeated...

Origami: The Japanese art of origami involves the repeated folding of a single piece of paper to create various art forms. When the upper right corner of a rectangular $21.6-\mathrm{cm}$ by $28-\mathrm{cm}$ piece of paper is folded down until the corner is flush with the other side, the length $L$ of the fold is related to the angle $\theta$ by $L=\frac{10.8}{\sin \theta \cos ^{2} \theta} .$ (a) Show this is equivalent to $L=\frac{21.6 \sec \theta}{\sin (2 \theta)},(b)$ find the length of the fold if $\theta=30^{\circ},$ and (c) find the angle $\theta$ if $L=28.8 \mathrm{cm}$

Key Concepts

Trigonometric Identities

Trigonometric identities are formulas that express relationships between the trigonometric functions and allow one to transform and simplify expressions. For example, knowing that sin(2?) = 2 sin ? cos ? or that sec ? is the reciprocal of cos ? is fundamental in rewriting and understanding trigonometric equations. These identities are crucial when transforming one form of an expression into another, as well as when solving equations involving angles.

Algebraic Manipulation

Algebraic manipulation involves rewriting mathematical expressions using techniques such as factoring, substitution, and the use of identities. In problems that involve trigonometric forms, this skill enables one to show the equivalence of different representations of an expression. It is essential for simplifying complex expressions and making problems more tractable, whether for proofs or for finding numerical values.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the value of an angle or variable that satisfies a given trigonometric relationship. This process often requires the application of inverse trigonometric functions, careful consideration of the domain and periodicity of the trigonometric functions, and sometimes numerical methods when the equations are transcendental. It is a key skill for translating geometric configurations into quantitative solutions.

Geometric Interpretation of Trigonometry

The geometric interpretation of trigonometry ties the relationships between angles and side lengths directly to figures and shapes. By understanding how trigonometric functions model relationships in geometric constructs, one can analyze problems involving folding, rotation, or distances within shapes. This connection is particularly useful in applications where physical configuration and design are modeled mathematically.

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