The value of tan (tangent) is the ratio of the opposite side to the adjacent side in a right triangle. It can also be expressed as tan θ = sin θ / cos θ. Tangent is periodic with a period of 180° or π radians and is undefined for angles where cos θ = 0.
Class 12 Maths Chapter 2 Inverse Trigonometric Functions focuses on the inverse of trigonometric functions like sine and cosine and tangent or cosecant and secant and cotangent. It helps in finding angles when the value of a trigonometric function is given. The chapter covers domains ranges graphs and solving equations of inverse trigonometric functions.
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We are given:
tan⁻¹(tan 5π/6) + cos⁻¹(cos 13π/6)
Step 1: Simplify tan⁻¹(tan 5π/6)
The range of tan⁻¹ x is (-π/2, π/2).
For any angle θ, tan⁻¹(tan θ) gives the principal value of θ, which must lie in this range.
The angle 5π/6 lies outside this range. To bring it into the principal range, we use the periodicity of tan and adjust it:
5π/6 – π = -π/6
Thus:
tan⁻¹(tan 5π/6) = -π/6
Step 2: Simplify cos⁻¹(cos 13π/6)
The range of cos⁻¹ x is [0, π].
For any angle θ, cos⁻¹(cos θ) gives the principal value of θ, which must lie in this range.
The angle 13π/6 is outside this range. To bring it into the range, subtract 2π:
13π/6 – 2π = π/6
Thus:
cos⁻¹(cos 13π/6) = π/6
Step 3: Add the two results
Now, add the simplified terms:
tan⁻¹(tan 5π/6) + cos⁻¹(cos 13π/6) = -π/6 + π/6 = 0
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