Ch. 2 - Acute Angles and Right Triangles

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problema 5

Capitolo 3, Problema 5

CONCEPT PREVIEW Match each trigonometric function in Column I with its value in Column II. Choices may be used once, more than once, or not at all.
csc 60°

Guida verificata passo dopo passo

Step 1: Recall the definition of the cosecant function. Cosecant is the reciprocal of sine, so \(\csc \theta = \frac{1}{\sin \theta}\).

Step 2: Find the value of \(\sin 60^\circ\). From the special angles in trigonometry, \(\sin 60^\circ = \frac{\sqrt{3}}{2}\).

Step 3: Calculate \(\csc 60^\circ\) using the reciprocal relationship: \(\csc 60^\circ = \frac{1}{\sin 60^\circ} = \frac{1}{\frac{\sqrt{3}}{2}}\).

Step 4: Simplify the expression for \(\csc 60^\circ\) by multiplying numerator and denominator appropriately: \(\csc 60^\circ = \frac{2}{\sqrt{3}}\).

Step 5: Rationalize the denominator if needed by multiplying numerator and denominator by \(\sqrt{3}\) to get \(\csc 60^\circ = \frac{2\sqrt{3}}{3}\), which matches one of the values in Column II.

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Basic Trigonometric Ratios

Trigonometric functions such as sine, cosine, and cosecant relate the angles of a right triangle to the ratios of its sides. For example, csc θ is the reciprocal of sin θ, meaning csc θ = 1/sin θ. Understanding these ratios is essential for matching functions to their numerical values.

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Introduction to Trigonometric Functions

Special Angles and Their Values

Certain angles like 30°, 45°, and 60° have well-known exact trigonometric values involving square roots and fractions. For instance, sin 60° = √3/2 and csc 60° = 2/√3. Memorizing these special angle values helps quickly identify correct matches in problems.

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45-45-90 Triangles

Reciprocal Identities

Reciprocal identities express functions like cosecant, secant, and cotangent as reciprocals of sine, cosine, and tangent respectively. For example, csc θ = 1/sin θ. Recognizing these identities allows conversion between functions and their values, aiding in matching tasks.

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Pythagorean Identities

Pratica correlata

Domanda del libro di testo

CONCEPT PREVIEW Match each equation in Column I with the appropriate right triangle in Column II. In each case, the goal is to find the value of x.

x = 5 tan 38°

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Domanda del libro di testo

Match each trigonometric function in Column I with its value in Column II. Choices may be used once, more than once, or not at all.

cot 30°

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Domanda del libro di testo

Concept Check Match each angle in Column I with its reference angle in Column II. Choices may be used once, more than once, or not at all. See Example 1. I. II. 5. A. 45° 6. 212° B. 60° 7. C. 82° 8. D. 30° 9. E. 38° 10. F. 32°

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Domanda del libro di testo

CONCEPT PREVIEW Match each trigonometric function value or angle in Column I with its appropriate approximation in Column II.

Column I: 1.

sin⁻¹ 0.30

Column II:

A. 88.09084757°

B. 63.25631605°

C. 1.909152433°

D. 17.45760312°

E. 0.2867453858