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Ch. 1 - Angles and the Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 1 - Angles and the Trigonometric FunctionsProblema 1.2.67
Capitolo 1, Problema 1.2.67

In Exercises 63–68, find the exact value of each expression. Do not use a calculator. csc 37° sec 53° - tan 53° cot 37°

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Recall the complementary angle relationships: since 37° and 53° add up to 90°, we have \( \sin 37^\circ = \cos 53^\circ \) and \( \cos 37^\circ = \sin 53^\circ \). This will help simplify the trigonometric expressions.
Rewrite each trigonometric function in terms of sine and cosine: \( \csc 37^\circ = \frac{1}{\sin 37^\circ} \), \( \sec 53^\circ = \frac{1}{\cos 53^\circ} \), \( \tan 53^\circ = \frac{\sin 53^\circ}{\cos 53^\circ} \), and \( \cot 37^\circ = \frac{\cos 37^\circ}{\sin 37^\circ} \).
Substitute these into the expression: \( \csc 37^\circ \sec 53^\circ - \tan 53^\circ \cot 37^\circ = \frac{1}{\sin 37^\circ} \times \frac{1}{\cos 53^\circ} - \frac{\sin 53^\circ}{\cos 53^\circ} \times \frac{\cos 37^\circ}{\sin 37^\circ} \).
Use the complementary angle identities to replace \( \cos 53^\circ \) with \( \sin 37^\circ \) and \( \sin 53^\circ \) with \( \cos 37^\circ \), simplifying the expression to terms involving only \( \sin 37^\circ \) and \( \cos 37^\circ \).
Simplify the resulting expression by combining fractions and canceling common terms to find the exact value without using a calculator.

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Reciprocal Trigonometric Functions

Cosecant (csc) and secant (sec) are the reciprocals of sine and cosine, respectively. Specifically, csc θ = 1/sin θ and sec θ = 1/cos θ. Understanding these relationships helps simplify expressions involving these functions without a calculator.
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Introduction to Trigonometric Functions

Complementary Angle Relationships

Angles like 37° and 53° are complementary because they add up to 90°. Trigonometric functions of complementary angles are related, for example, sin(37°) = cos(53°) and tan(37°) = cot(53°). This property is useful for rewriting and simplifying expressions.
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Intro to Complementary & Supplementary Angles

Simplification of Trigonometric Expressions

Simplifying expressions involves substituting reciprocal and complementary angle identities and combining terms. Recognizing patterns and using fundamental identities allows finding exact values without a calculator, which is essential for solving the given problem.
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Simplifying Trig Expressions
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