Solve each equation on the interval 0 ≤θ<2 π3-sinθ=cos(2 θ)...

Name: Problem 77, Chapter 8: Algebra and Trigonometry
Uploaded: 2020-06-19T02:29:23-08:00
Duration: 2 min 16 s
Channel: Suman Saurav Thakur
Description: Solve each equation on the interval 0 ≤θ<2 π3-sinθ=cos(2 θ)

Key Concepts

Trigonometric Functions and Their Ranges

This concept involves understanding the inherent properties of sine and cosine functions, including their domains and ranges. Recognizing that the sine function outputs values between -1 and 1, while the cosine function (even when its argument is doubled) also remains within this same range, is critical when equating expressions that contain these functions. Such an understanding helps establish whether an equation might have potential solutions or if a contradiction exists based on the range of the involved functions.

Double Angle Identity

The double angle identity is a fundamental tool in trigonometry that allows one to express trigonometric functions of double angles in terms of single angles. In the context of solving equations like the one given, expressing cos(2?) using its equivalent forms (such as 1 - 2sin²?, 2cos²? - 1, or cos²? - sin²?) can simplify the equation and make it easier to compare or combine with other trigonometric expressions. This identity is central to transforming the problem into a solvable form.

Equation Transformation in Trigonometry

This key concept involves rewriting a trigonometric equation into an equivalent form that is easier to analyze or solve. Typical techniques include applying known identities, isolating trigonometric functions, and sometimes converting all terms to the same function to facilitate solving. Transformation is essential in handling complex trigonometric equations by reducing them to simpler algebraic or standard trigonometric forms.

Periodic Properties of Trigonometric Functions

Trigonometric functions are periodic in nature, meaning their values repeat over specific intervals. When solving trigonometric equations within a defined interval, it is important to consider the periodic behavior to identify all possible solutions within that interval. This concept ensures that the full set of solutions, if any exist, is correctly captured despite the repetitive nature of the trigonometric functions.

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