Ch. 5 - Trigonometric Identities

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

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problema 5.1.12

Capitolo 6, Problema 5.1.12

Find sin θ.
cos θ = 5/6, θ in quadrant I

Guida verificata passo dopo passo

Recall the Pythagorean identity: \(\sin^{2} \theta + \cos^{2} \theta = 1\).

Substitute the given value of \(\cos \theta = \frac{5}{6}\) into the identity: \(\sin^{2} \theta + \left(\frac{5}{6}\right)^{2} = 1\).

Calculate \(\left(\frac{5}{6}\right)^{2}\) which is \(\frac{25}{36}\), so the equation becomes \(\sin^{2} \theta + \frac{25}{36} = 1\).

Isolate \(\sin^{2} \theta\) by subtracting \(\frac{25}{36}\) from both sides: \(\sin^{2} \theta = 1 - \frac{25}{36}\).

Since \(\theta\) is in quadrant I, where sine is positive, take the positive square root: \(\sin \theta = \sqrt{1 - \left(\frac{5}{6}\right)^{2}}\).

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Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1 for any angle θ. This relationship allows you to find the sine of an angle if the cosine is known, by rearranging the formula to sin θ = ±√(1 - cos²θ).

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6:25

Pythagorean Identities

Sign of Trigonometric Functions in Quadrants

The sign of sine and cosine depends on the quadrant where the angle lies. In quadrant I, both sine and cosine values are positive, which helps determine the correct sign when calculating sin θ from the Pythagorean identity.

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6:36

Quadratic Formula

Using Given Cosine Value to Find Sine

Given cos θ = 5/6, you substitute this value into the Pythagorean identity to find sin θ. Since θ is in quadrant I, sin θ will be positive, so sin θ = √(1 - (5/6)²) = √(1 - 25/36) = √(11/36) = √11/6.

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5:08

Sine, Cosine, & Tangent of 30°, 45°, & 60°

Pratica correlata

Domanda del libro di testo

The half-angle identity

tan A/2 = ± √[(1 - cosA)/(1 + cos A)]

can be used to find tan 22.5° = √(3 - 2√2), and the half-angle identity

tan A/2 = sin A/(1 + cos A)

can be used to find tan 22.5° = √2 - 1. Show that these answers are the same, without using a calculator. (Hint: If a > 0 and b > 0 and a² = b², then a = b.)

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Domanda del libro di testo

Verify that each equation is an identity.

(sin⁴ α - cos⁴ α )/(sin² α - cos² α) = 1

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Domanda del libro di testo

For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.

cos² x

A. sin ^2 x/cos ^2 x

B.1/(sec ^2 x)