MATH SOLVE

3 months ago

Q:
# plsss help me simplify these...and guys plz don't delete my questions ....

Accepted Solution

A:

Answer:(i) sin⁻¹ [ x / a ](iii) tan⁻¹ [ tan(π/4 − x) ] = π/4 − x + kπ {-π/4 + kπ ≤ x ≤ 3π/4 + kπ}(v) tan⁻¹ [ √(1 − x²) / √(1 + x²) ] + π/4Step-by-step explanation:(i) tan⁻¹ [ x / √(a² − x²) ]Draw a right triangle, where x is one of the sides and a is the hypotenuse. Using Pythagorean theorem, the other side is √(a² − x²). If we say the angle opposite of the x side is θ, then:θ = tan⁻¹ [ x / √(a² − x²) ]But we can also define θ using sine:θ = sin⁻¹ [ x / a ](iii) tan⁻¹ [ (cos x − sin x) / (cos x + sin x) ]Factor cos x from numerator and denominator:tan⁻¹ [ cos x (1 − tan x) / (cos x (1 + tan x)) ]tan⁻¹ [ (1 − tan x) / (1 + tan x) ]Using angle sum formula:tan⁻¹ [ 1 / tan(x + π/4) ]tan⁻¹ [ cot(x + π/4) ]Using phase shift:tan⁻¹ [ tan(π/2 − (x + π/4)) ]tan⁻¹ [ tan(π/2 − x − π/4) ]tan⁻¹ [ tan(π/4 − x) ]Or:π/4 − x + kπ {-π/4 + kπ ≤ x ≤ 3π/4 + kπ}(v) tan⁻¹ [ (√(1 + x²) + √(1 − x²)) / (√(1 + x²) − √(1 − x²)) ]Notice that (√(1 + x²))² + (√(1 − x²))² = 2.Therefore, we can substitute √(1 + x²) = √2 cos u, and √(1 − x²) = √2 sin u.tan⁻¹ [ (√2 cos u + √2 sin u) / (√2 cos u − √2 sin u) ]tan⁻¹ [ (cos u + sin u) / (cos u − sin u) ]Notice this is the inverse of what we had in problem (iii). Therefore, using the same logic:tan⁻¹ [ tan(u + π/4) ]u + π/4Substituting back:= sin⁻¹ [ √(1 − x²) / √2 ] + π/4Or:= cos⁻¹ [ √(1 + x²) / √2 ] + π/4Or:= tan⁻¹ [ √(1 − x²) / √(1 + x²) ] + π/4