University:
North Central College
Course:
MATH 152 | Calculus II
Academic year:

2024
Views:

94

Pages:

2
Author:

nesikhiyawl0

Question 1 Tell, what is the derivative of sec (x)? Answer 1 sec (x) = 1 / cos(x) d/dx (1/cos(x)) = d/dx (cos(x)⁻¹) = -cos(x)⁻² d/dx cos(x) = -1/cos(x)² * (-sin(x)) = sin(x) / cos(x)² Answer 2 It is sin(x) / cos(x)² You're welcome. Question 2 Find the value of arcsin (1) Answer 1 Within the range [0, π/2] the only value for θ that satisfies this condition is θ = π/2 Thus, arcsin (1) = π/2 Answer 2 arcsin (1) = π/2 Question 3 Prove that cos(0)=1 Answer 1 cos(x) = adjacent side / hypotenuse When x = 0, adjacent side length = hypotenuse length. Thus, cos 0 = 1 Answer 2 Use the Unit Circle. At 0 degrees, the angle intercepts at the coordinate (1,0) and x is the cos value and y is the sin value. Question 4 How do you simplify cos (sin⁻¹x)? Answer 1 After using u = arcsin(x) or x = sin(u) transforms cos (arcsin x) = cos u = √(1 - (sin u)²) = √(1 - x²) Answer 2 cos(sin⁻¹x) Let sin⁻¹x = θ Then x = sinθ Place the value for x in cos(sin⁻¹x) ⇒ cos(sin⁻¹(sinθ)) The resulting equation is, ⇒ cosθ We have sin²θ + cos²θ = 1 ⇒ cos²θ = 1 - sin²θ ⇒ cosθ = √(1 - sin²θ) We have already found that x = sinθ, then x² = sin²θ. Now put x² in the place for sin²θ cosθ = √(1 - x²) Question 5 Find the value of sin60. Answer 1 Consider a triangle and cut a slice of it From the Pythagorean theorem a² = h² + (a²/4) h² = a² - a²/4 h² = (3a²)/4 So, h = a√(3/4) = a√3/2 Now, from trigonometry: h = a sin 60 a√3/2 = a sin 60 sin 60 = √3/2 Answer 2 sin 60 = √3/2