Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Algebra of Trigonometry**
**Double-angle Formulas**
1. \(\sin 2\theta = 2 \sin \theta \cos \theta\)
2. \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 1 - 2 \sin^2 \theta = 2 \cos^2 \theta - 1\)
3. \(\tan 2\theta = \frac{2 \tan^2 \theta}{1 - \tan^2 \theta}\)
4. Derive the formula of \(\cos 4\theta\)
The provided text includes several common double-angle trigonometric formulas, which are important for simplifying and solving trigonometric equations. Let's go through the details of each formula:
1. **Sine Double Angle Formula:**
\[\sin 2\theta = 2 \sin \theta \cos \theta\]
This formula expresses the sine of double an angle in terms of the sine and cosine of the angle.
2. **Cosine Double Angle Formulas:**
\[\cos 2\theta = \cos^2 \theta - \sin^2 \theta\]
This primary formula for \(\cos 2\theta\) can be further expanded into two equivalent forms depending on the known values:
\[\cos 2\theta = 1 - 2 \sin^2 \theta\]
\[\cos 2\theta = 2 \cos^2 \theta - 1\]
These various forms can be used based on which trigonometric values are more convenient or known in a problem.
3. **Tangent Double Angle Formula:**
\[\tan 2\theta = \frac{2 \tan^2 \theta}{1 - \tan^2 \theta}\]
This formula provides the tangent of double an angle in terms of the tangent of the angle. It is particularly useful when working with angles derived from tangent functions.
4. **Deriving the Formula for \(\cos 4\theta\):**
This step prompts the derivation of the formula for the cosine of four times an angle (\(4\theta\)) based on the given double-angle formulas. This derivation process involves using the double-angle formula for cosine repeatedly to express \(\cos 4\theta\) in terms of \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fedc716be-e066-4234-bd1a-1a86837575c3%2Fa90532ee-3bd1-438d-b370-f3b383f139a5%2Fkar7vx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Algebra of Trigonometry**
**Double-angle Formulas**
1. \(\sin 2\theta = 2 \sin \theta \cos \theta\)
2. \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 1 - 2 \sin^2 \theta = 2 \cos^2 \theta - 1\)
3. \(\tan 2\theta = \frac{2 \tan^2 \theta}{1 - \tan^2 \theta}\)
4. Derive the formula of \(\cos 4\theta\)
The provided text includes several common double-angle trigonometric formulas, which are important for simplifying and solving trigonometric equations. Let's go through the details of each formula:
1. **Sine Double Angle Formula:**
\[\sin 2\theta = 2 \sin \theta \cos \theta\]
This formula expresses the sine of double an angle in terms of the sine and cosine of the angle.
2. **Cosine Double Angle Formulas:**
\[\cos 2\theta = \cos^2 \theta - \sin^2 \theta\]
This primary formula for \(\cos 2\theta\) can be further expanded into two equivalent forms depending on the known values:
\[\cos 2\theta = 1 - 2 \sin^2 \theta\]
\[\cos 2\theta = 2 \cos^2 \theta - 1\]
These various forms can be used based on which trigonometric values are more convenient or known in a problem.
3. **Tangent Double Angle Formula:**
\[\tan 2\theta = \frac{2 \tan^2 \theta}{1 - \tan^2 \theta}\]
This formula provides the tangent of double an angle in terms of the tangent of the angle. It is particularly useful when working with angles derived from tangent functions.
4. **Deriving the Formula for \(\cos 4\theta\):**
This step prompts the derivation of the formula for the cosine of four times an angle (\(4\theta\)) based on the given double-angle formulas. This derivation process involves using the double-angle formula for cosine repeatedly to express \(\cos 4\theta\) in terms of \
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