SUMMARY
The expression 8 cos² x - 6 sin x cos x + 2 can be rewritten in the form rcos(2x + α) + s. The constants are determined as r = 10, s = 2, and α = arctan(3/4). This transformation utilizes trigonometric identities such as cos 2x = cos² x - sin² x and sin 2α = 2 sin x cos x to simplify the expression effectively. The maximum and minimum values of the expression are found to be 12 and 2, respectively, as x varies.
PREREQUISITES
- Understanding of trigonometric identities, specifically cos 2x and sin 2α.
- Familiarity with the transformation of trigonometric expressions into the form rcos(x + α).
- Knowledge of how to find maximum and minimum values of trigonometric functions.
- Basic algebraic manipulation skills to simplify expressions.
NEXT STEPS
- Study the derivation of rcos(x + α) from linear combinations of sine and cosine.
- Learn about the application of trigonometric identities in simplifying complex expressions.
- Explore methods for determining extrema of trigonometric functions.
- Investigate the use of the arctangent function in finding angles from ratios.
USEFUL FOR
Students studying trigonometry, mathematics educators, and anyone interested in mastering the transformation of trigonometric expressions for problem-solving in calculus and algebra.
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