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 2.3.60

Capitolo 3, Problema 2.3.60

Use a calculator to determine whether each statement is true or false. A true statement may lead to results that differ in the last decimal place due to rounding error. tan² 72°25' + 1 = sec² 72°25'

Guida verificata passo dopo passo

First, recognize the Pythagorean identity involving tangent and secant: \(\tan^2 \theta + 1 = \sec^2 \theta\). This identity holds true for any angle \(\theta\) where these functions are defined.

Convert the angle from degrees and minutes to decimal degrees for calculator input. Since 1 minute is \(\frac{1}{60}\) degrees, calculate \(72°25' = 72 + \frac{25}{60}\) degrees.

Calculate \(\tan(72.4167°)\) using the calculator, then square the result to find \(\tan^2(72.4167°)\).

Add 1 to the value obtained in the previous step to compute \(\tan^2(72.4167°) + 1\).

Calculate \(\sec(72.4167°)\) by finding \(\frac{1}{\cos(72.4167°)}\) using the calculator, then square this value to get \(\sec^2(72.4167°)\). Finally, compare this result with the value from step 4 to determine if the statement is true or false, considering minor differences due to rounding.

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Pythagorean Identity for Tangent and Secant

The identity tan²θ + 1 = sec²θ is a fundamental trigonometric identity derived from the Pythagorean theorem. It relates the square of the tangent of an angle to the square of the secant of the same angle, holding true for all angles where these functions are defined.

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Percorso guidato

6:25

Pythagorean Identities

Angle Conversion and Notation

Angles given in degrees and minutes (e.g., 72°25') must be converted to decimal degrees or radians for calculator use. One minute equals 1/60 of a degree, so 72°25' equals 72 + 25/60 degrees, ensuring accurate input for trigonometric calculations.

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Percorso guidato

06:01

i & j Notation

Rounding Errors in Calculator Computations

Calculators approximate trigonometric values, which can cause minor differences in the last decimal places. Understanding that slight discrepancies do not invalidate identities is important when verifying trigonometric statements numerically.

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Percorso guidato

05:21

Computing the Cross Product

Pratica correlata

Domanda del libro di testo

CONCEPT PREVIEW Match the measure of bearing in Column I with the appropriate graph in Column II.

I. N 70° W

II. 1. A. B. C. 2. 3. 4. D. E. F. 5. N 70° W 6. 7. G. H. 8. 9. 10. I. J.

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

Column I:

cos⁻¹ 0.45

Column II:

A. 88.09084757°

B. 63.25631605°

C. 1.909152433°

D. 17.45760312°

E. 0.2867453858

F. 1.962610506

G. 14.47751219°

H. 1.015426612

I. 1.051462224

J. 0.9925461516

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Domanda del libro di testo

CONCEPT PREVIEW Match the measure of bearing in Column I with the appropriate graph in Column II.

I. 10. N 70° E

II. 1. A. B. C. 2. 3. 4. D. E. F. 5. 6. 7. G. H. 8. 9. I. J.

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Domanda del libro di testo

Use a calculator to evaluate each expression. sin 35° cos 55° + cos 35° sin 55°

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Domanda del libro di testo

Solve each problem. See Examples 3 and 4. The figure to the right indicates that the equation of a line passing through the point (a, 0) and making an angle θ with the x-axis is y = (tan θ) (x - a). Find an equation of the line passing through the point (5, 0) that makes an angle of 15° with the x-axis.

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Use a calculator to determine whether each statement is true or false. A true statement may lead to results that differ in the last decimal place due to rounding error. ½ sin 40° = sin [½ (40°)]

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