The equation 3^sin2 x+2 cos^2 x+3^1-sin2 x+2 sin^1 x=28 is satis- fied for the values of x given

step 1 In this wrong, the given equation can be retaken as 3 to the power of sine 2x plus 2 cost square

The equation 3^sin2 x+2 cos^2 x+3^1-sin2 x+2 sin^1 x=28 is...

Key Concepts

Exponential Equations

This concept involves solving equations where the unknown variable appears in the exponent. The approach often includes rewriting the equation in a form where a substitution can turn an otherwise complicated equation into an algebraic one. Recognizing the structure of an exponential equation, especially when the exponents themselves are expressions involving other functions (like trigonometric functions), is key to applying methods such as taking logarithms or substituting variables for the exponent’s value.

Trigonometric Identities

Trigonometric identities, such as the double-angle formulas (e.g. sin 2x = 2 sin x cos x) and power reduction formulas (e.g. 2 cos^2 x = 1+cos 2x, 2 sin^2 x = 1-cos 2x), are crucial when simplifying and transforming expressions that involve trigonometric functions. Using these identities helps in rewriting complex trigonometric expressions into simpler forms, facilitating further algebraic manipulation and the eventual solution of the equation.

Solving Trigonometric Equations

Once trigonometric expressions are simplified using identities, the next step typically involves solving the resulting trigonometric equation. This process may include rewriting sums of sine and cosine functions as a single sine (or cosine) function with an amplitude and phase shift and then applying inverse trigonometric functions together with an understanding of the periodicity of these functions. This ensures that all solutions within the domain are correctly identified.

Range Analysis and Extraneous Solutions

When solving equations that involve trigonometric and exponential functions, it is important to analyze the possible range and domain of the expressions involved. Understanding the maximum and minimum possible values that a trigonometric expression can attain helps in eliminating impossible scenarios or extraneous solutions that might arise during algebraic manipulation. This step guarantees that the solutions found are valid within the constraints of the functions.

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