뒤로Analytic Trigonometry: Double-Angle, Power-Reducing, and Half-Angle Formulas
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Analytic Trigonometry
Double-Angle, Power-Reducing, and Half-Angle Formulas
This section explores advanced trigonometric identities used to simplify expressions and solve equations. The focus is on double-angle, power-reducing, and half-angle formulas, which are essential tools in precalculus for manipulating trigonometric functions.
Double-Angle Formulas
Double-angle formulas express trigonometric functions of twice an angle in terms of functions of the original angle. These are useful for finding exact values and simplifying expressions.
Sine Double-Angle Formula:
Cosine Double-Angle Formulas:
Tangent Double-Angle Formula:
Example: If lies in quadrant II and , find using the double-angle formula.
Verifying Trigonometric Identities
Trigonometric identities can be verified by algebraic manipulation, substitution, and using known formulas. Double-angle formulas are often used in these verifications.
Key Point: To verify an identity, rewrite one or both sides using double-angle formulas and simplify.
Example: Verify by expanding and .
Power-Reducing Formulas
Power-reducing formulas express powers of sine and cosine in terms of the first power, making them useful for integration and simplification.
Example: Write as an equivalent expression without powers greater than 1 using the power-reducing formula.
Half-Angle Formulas
Half-angle formulas allow calculation of trigonometric functions of half an angle. The sign of the result depends on the quadrant in which the half-angle lies.
Key Point: The symbol does not mean there are two possible values; it indicates the sign must be determined based on the quadrant.
Example: Use the half-angle formula to find the exact value of given and the quadrant.
Half-Angle Formulas for Tangent
Half-angle formulas for tangent are derived from the sine and cosine half-angle formulas:
