General Solution of Equation tanθ = tanα General Solution of Equation: Given tanθ = tanα , sin θ cosα = sinα cosθ, sin θ cosα – sinα cosθ = 0, , sin (θ – α) = 0 , θ – α = Sin⁻¹(0), , , Note: For general solution of the equation tanθ = k, Read more about General Solution of Equation tanθ = tanα[…]

General Solution of Equation cosθ = cosα General Solution of Equation cosθ = cosα: Given cosθ = cosα cosθ – cosα = 0 , cosθ – cosα = , , ,  and , Case(i): , , , . Case(ii): . , , , From case i and ii , Note: For general solution of the Read more about General Solution of Equation cosθ = cosα[…]

Trigonometric Ratios for Compound Angles – Part2 Cosine of the Sum of Two Angles:  for all angles A and B Proof: Let X’OX and YOY’ be the coordinate axes. Consider a unit circle with O as the center. Let P₁, P₂ and P₃ be the three points on the circle such that   We know Read more about Trigonometric Ratios for Compound Angles – Part2[…]

Trigonometric Ratios for Compound Angles – Part1 Cosine of the Difference of Two Angles:  for all angles A and B Proof: Let X’OX and YOY’ be the coordinate axes. Consider a unit circle with O as the center. Let P₁, P₂ and P₃ be the three points on the circle such that   We know Read more about Trigonometric Ratios for Compound Angles – Part1[…]

Elementary Equations – Problems 1.Find the values of θ which satisfy rsinθ = 3 and r = 4 (1 + sinθ), 0 ≤ θ ≤ 2π. solution: rsinθ = 3 and r = 4 (1 + sinθ), 0 ≤ θ ≤ 2π rsinθ = 3 r = 3/sinθ 3/sinθ = 4 (1 + sinθ) 3 Read more about Elementary Equations – Problems[…]

General Solution of Some Standard Equations General solution of Equation sinθ = sinα:  sinθ = sinα  sinθ – sinα = 0, , , , , , θ + α = (2m + 1) π θ = (2m + 1) π – α, m ϵ Z ….(1) (θ – α)/2 = sin⁻¹(0) (θ – α)/2 = Read more about General Solution of Some Standard Equations[…]

 Fundamental Laws of Logarithms- Part3 7. For a, n > 0 and a ≠ 1; Proof: Let . Then . Therefore, , Examples: (1) (2) (3) 8. , where a, n > 0, a ≠ 1 Proof: Let then, , , , , qx  = yp, ⇒  x = (p/q)y, ⇒ . Example: What is Read more about Fundamental Laws of Logarithms- Part3[…]

Fundamental Laws of Logarithms – Part 2 3. For m, n, a > 0, a≠1; Proof: let , , , , , Hence Proved. 4. For a > 1, if Proof: , , ,  (since a > 1), Hence proved. 5. For 0 < a < 1, if Proof: , , , , (since 0 Read more about Fundamental Laws of Logarithms – Part 2[…]

Fundamental Laws of Logarithms – Part 1 1.For m, n > 0, a ≠ 1; Proof: Let  and , …(1), ….(2), From equation (1) and (2), , , , , Hence proved. In general, if are positive real numbers, then, , 2. For m, n, a>0, a ≠ 1; . Proof: Let and …(1), ….(2), Read more about Fundamental Laws of Logarithms – Part 1[…]