Inverse Trigonometric Functions

Answer

Step 1: Let sin⁻¹(1/2) = y, so sin y = 1/2 Step 2: The range of principal values is [-π/2, π/2] Step 3: Find y in this range where sin y = 1/2 Step 4: sin(π/6) = 1/2 and π/6 ∈ [-π/2, π/2] Answer: π/6 …

Answer

Step 1: Let cos⁻¹(-1/2) = y, so cos y = -1/2 Step 2: The range of principal values is [0, π] Step 3: Find y in this range where cos y = -1/2 Step 4: cos(2π/3) = cos(π - π/3) = -cos(π/3) = -1/2 Step 5:…

Answer

Step 1: Check if 5π/6 is in the principal range [-π/2, π/2] Step 2: 5π/6 ∉ [-π/2, π/2], so we need to find an equivalent angle Step 3: sin(5π/6) = sin(π - π/6) = sin(π/6) = 1/2 Step 4: Now find x wher…

Answer

Step 1: Check if 7π/6 is in the principal range [0, π] Step 2: 7π/6 ∉ [0, π] (since 7π/6 > π), so find an equivalent angle Step 3: 7π/6 = 2π - 5π/6, so cos(7π/6) = cos(2π - 5π/6) = cos(-5π/6) = cos(5π…

Answer

Use this formula when: • You need to find the sum of inverse sine and cosine of the same value • You have expressions like sin⁻¹(3/5) + cos⁻¹(3/5) • Domain: x ∈ [-1, 1] Example: Find sin⁻¹(3/5) + cos…

Answer

Step 1: Use the formula: tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1-xy)) when xy < 1 Step 2: Check condition: xy = 2(3) = 6 > 1, so formula gives tan⁻¹(sum/(1-product)) Step 3: For xy > 1 (both positive): tan⁻¹…

Answer

Step 1: For sin⁻¹ to be defined, we need: -1 ≤ (2-3x²) ≤ 1 Step 2: From -1 ≤ 2-3x²: -1 ≤ 2-3x² -3x² ≤ -3 x² ≥ 1 |x| ≥ 1, so x ≤ -1 or x ≥ 1 Step 3: From 2-3x² ≤ 1: 2-3x² ≤ 1 -3x² ≤ …

Answer

Step 1: Let θ = sin⁻¹x, so sin θ = x and θ ∈ (-π/2, π/2) Step 2: In this range, cos θ > 0, so: cos θ = √(1 - sin²θ) = √(1 - x²) Step 3: Therefore: tan θ = sin θ / cos θ = x / √(1-x²) Step 4: S…