This chapter focuses on inverse trigonometric functions and their properties. Understanding these functions is crucial for solving equations and integrals in calculus.
Inverse Trigonometric Functions – Formula & Equation Sheet
Essential formulas and equations from Mathematics Part - I, tailored for Class 12 in Mathematics.
This one-pager compiles key formulas and equations from the Inverse Trigonometric Functions chapter of Mathematics Part - I. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
sin⁻¹(x): [-1, 1] → [-π/2, π/2]
sin⁻¹(x) defines the inverse sine function with domain [-1, 1] and range [-π/2, π/2], used to determine angles whose sine is x.
cos⁻¹(x): [-1, 1] → [0, π]
cos⁻¹(x) defines the inverse cosine function with domain [-1, 1] and range [0, π], used to find angles whose cosine is x.
tan⁻¹(x): (-∞, ∞) → (-π/2, π/2)
tan⁻¹(x) defines the inverse tangent function with domain (-∞, ∞) and range (-π/2, π/2), used to find angles whose tangent is x.
cosec⁻¹(x): |x| ≥ 1 → [-π/2, -π/2) ∪ (0, π/2)
cosec⁻¹(x) defines the inverse cosecant function with domain |x| ≥ 1, used to find angles whose cosecant is x.
sec⁻¹(x): |x| ≥ 1 → [0, π/2) ∪ (π/2, π]
sec⁻¹(x) defines the inverse secant function with domain |x| ≥ 1, used to find angles whose secant is x.
cot⁻¹(x): (-∞, ∞) → (0, π)
cot⁻¹(x) defines the inverse cotangent function with domain (-∞, ∞) and range (0, π), used to find angles whose cotangent is x.
sin(sin⁻¹(x)) = x, for x ∈ [-1, 1]
Confirms that applying sine to its inverse sine function returns the original input x within the defined domain.
cos(cos⁻¹(x)) = x, for x ∈ [-1, 1]
Confirms that applying cosine to its inverse cosine function returns the original input x within the defined domain.
tan(tan⁻¹(x)) = x, for x ∈ R
Confirms that applying tangent to its inverse tangent function returns the original input x for any real number.
cosec(cosec⁻¹(x)) = x, for |x| ≥ 1
Confirms that applying cosecant to its inverse cosecant function returns x for any value satisfying the amplitude restriction.
sec(sec⁻¹(x)) = x, for |x| ≥ 1
Confirms that applying secant to its inverse secant function returns x for values satisfying domain restrictions.
cot(cot⁻¹(x)) = x, for x ∈ R
Confirms that applying cotangent to its inverse cotangent function returns the original input x for any real number.
Equations
sin(-x) = -sin(x)
This equation reflects the odd property of the sine function, demonstrating that it is symmetric about the origin.
cos(-x) = cos(x)
This equation reflects the even property of the cosine function, indicating symmetry about the y-axis.
tan(-x) = -tan(x)
This equation shows the odd property of the tangent function, illustrating its parity symmetry.
sin²(x) + cos²(x) = 1
The Pythagorean identity that relates the sine and cosine functions of the same angle.
1 + tan²(x) = sec²(x)
This identity links tangent and secant, derived from the fundamental Pythagorean identity.
1 + cot²(x) = csc²(x)
This identity connects cotangent and cosecant functions, also derived from the fundamental Pythagorean identity.
cot⁻¹(x) + tan⁻¹(x) = π/2
This identity illustrates the relationship between inverse cotangent and inverse tangent functions.
sin⁻¹(x) + cos⁻¹(x) = π/2
This identity demonstrates the complementary relationship between inverse sine and cosine functions.
sin(π/2 - x) = cos(x)
Demonstrates the co-function identity linking sine and cosine for complementary angles.
tan(π/4) = 1
A special value indicating that at an angle of π/4, the tangent function equals 1.
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