Ch. 3 - Trigonometric Identities and Equations

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Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problema 25

Capitolo 3, Problema 25

Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). sin 2x = √3 / 2

Guida verificata passo dopo passo

Start by writing down the given equation: $\sin 2x = \frac{\sqrt{3}}{2}$.

Recall that $\sin \theta = \frac{\sqrt{3}}{2}$ at specific standard angles. Identify all angles $\theta$ in the interval $[0, 2\pi)$ where this is true. These angles are $\theta = \frac{\pi}{3}$ and $\theta = \frac{2\pi}{3}$.

Since the equation involves \$2x$, set $2x = \frac{\pi}{3} + 2k\pi$ and $2x = \frac{2\pi}{3} + 2k\pi$, where $k$ is any integer, to account for the periodicity of the sine function.

Solve each equation for $x$ by dividing both sides by 2: $x = \frac{\pi}{6} + k\pi$ and $x = \frac{\pi}{3} + k\pi$.

Find all values of $x$ within the interval $[0, 2\pi)$ by substituting integer values of $k$ (such as $k=0$ and $k=1$) into the expressions for $x$.

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Double-Angle Identities

Double-angle identities express trigonometric functions of multiples of an angle in terms of single angles. For example, sin(2x) = 2 sin x cos x. These identities help simplify and solve equations involving multiple angles like sin 2x.

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Percorso guidato

05:06

Double Angle Identities

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within a given interval. This often requires using inverse functions and considering the periodicity of sine, cosine, or tangent.

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Percorso guidato

4:34

How to Solve Linear Trigonometric Equations

Unit Circle and Interval Constraints

The unit circle helps visualize sine and cosine values for angles between 0 and 2π. Understanding the interval [0, 2π) is crucial to find all valid solutions within one full rotation, considering the periodic nature of trigonometric functions.

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Percorso guidato

06:11

Introduction to the Unit Circle

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