Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It involves functions such as sine, cosine, tangent, and their reciprocals. Trigonometry is widely used in geometry, physics, engineering, and astronomy for solving problems related to angles, distances, and wave patterns.
Class 12 Maths Chapter 2 Inverse Trigonometric Functions deals with the inverse of trigonometric functions like sine and cosine and tangent or cosecant and secant and cotangent. It helps find angles from given function values. The chapter covers domains and ranges graphs and solving equations involving inverse trigonometric functions.
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We are given:
cos⁻¹ α + cos⁻¹ β + cos⁻¹ γ = 3π
Step 1: Using the property of inverse cosine
The principal value range of cos⁻¹ x is [0, π].
For the sum of three inverse cosine terms to equal 3π, each term must be equal to π. This means:
cos⁻¹ α = π, cos⁻¹ β = π, cos⁻¹ γ = π
Step 2: Find the values of α, β, and γ
When cos⁻¹ α = π, then cos π = -1, so:
α = -1, β = -1, γ = -1
Step 3: Substitute into the expression
We are given to calculate:
α(β + γ) + β(γ + α) + γ(α + β)
Replace α = -1, β = -1, and γ = -1:
(-1)((-1) + (-1)) + (-1)((-1) + (-1)) + (-1)((-1) + (-1))
Simplify each term:
(-1)(-2) + (-1)(-2) + (-1)(-2) = 2 + 2 + 2 = 6
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