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The trigonometric identity \(\cos\left(\frac{3\pi}{2} + A\right) = \sin(A)\) demonstrates the phase shift property of cosine function. When we add \(3\pi/2\) (270 degrees) to angle A, the cosine function becomes equivalent to the sine of angle A.
The calculator uses the trigonometric identity:
Where:
Explanation: This identity shows the phase relationship between cosine and sine functions with a 3π/2 phase shift.
Details: The identity \(\cos\left(\frac{3\pi}{2} + A\right) = \sin(A)\) is derived from the cosine addition formula and the known values of trigonometric functions at special angles. It represents a phase shift of 270 degrees in the cosine function.
Tips: Enter the angle A in radians. The calculator will compute \(\cos\left(\frac{3\pi}{2} + A\right)\) using the equivalent \(\sin(A)\) identity.
Q1: Why does cos(3π/2 + A) equal sin(A)?
A: This is due to the phase shift properties of trigonometric functions. Adding 3π/2 (270°) to the angle in cosine function produces the same result as the sine function of the original angle.
Q2: Can I use degrees instead of radians?
A: The calculator requires input in radians. Convert degrees to radians by multiplying by π/180 before entering the value.
Q3: What are the limitations of this identity?
A: This identity holds true for all real values of angle A and is a fundamental trigonometric identity with no limitations.
Q4: How is this identity useful in practice?
A: This identity is useful in simplifying trigonometric expressions, solving equations, and in various applications of trigonometry in physics and engineering.
Q5: Are there similar identities for other phase shifts?
A: Yes, there are multiple trigonometric identities for different phase shifts, such as cos(π/2 + A) = -sin(A), cos(π + A) = -cos(A), etc.