Trigonometric Functions #11 - Questions and Answers

Question 1 What is the derivative of y = tan(x)? Answer 1 The derivative of tan x is sec²x To see why, you'll need to know a few results. First, you need to know that the derivative of sin x is cos x. Once all those pieces are in place, the differentiation goes as follows: d/dx tan x = d/dx (sin x / cos x) = (cos x d/dx sin x - sin x d/dx cos x) / cos²x (using Quotient Rule) = (cos x * cos x - sin x * (-sin x)) / cos²x = (cos²x + sin²x) / cos²x = 1 / cos²x (using the Pythagorean Identity) = sec²x Answer 2 Given, y = tan x. Let u = sinx and v = cos x On applying quotient rule on y = sinx / cosx, we get, dy/dx = (v * du/dx - u * dv/dx) / v² dy/dx = (cos x * cos x - sin x * (-sin x)) / cos²x dy/dx = (cos²x + sin²x) / cos²x dy/dx = 1 / cos²x dy/dx = sec²(x) Thus, the derivative of y = tan(x) is sec²(x) Answer 3 y = tan(x) Differentiate both sides of the equation. (y') = (tan(x))' The derivative of y with respect to x is y'. y' The derivative of tan(x) with respect to x is sec²(x) sec²(x) Reform the equation by setting the left side equal to the right side. y' = sec²(x) Replace y' with dy/dx dy/dx = sec²(x) Question 2 What is the derivative of sin²(x) Answer 1 y = sin²(x) First -method:- dy/dx = 2sinxcosx = sin 2x. Answer. Second method:- y = sin²(x). we know that.cos 2A = 1 - 2.sin²(A). or. sin²(A) = (1 - cos 2A)/2 y= (1 - cos 2x)/2 y= 1/2 - (1/2) cos 2x dy/dx = 0 - (1/2)(-sin 2x).2 dy/dx = sin 2x. Answer. Answer 2 y = sin² x. Let sin x = t, dt/dx = cos x So y = t². Diff. W.r.x. dy/dx = 2t * dt/dx => dy/dx = 2 sin x cos x Question 3 How do you find the value of tan 90? Answer 1 sin 90 = 1, cos 90 = 0 tan 90 = sin 90 / cos 90 = 1 / 0 = ∞ Answer 2 Since the angle is 90°, there will be a unit circle instead of a special triangle. Any point at 90° can be described as (0, y). However, to make things easier to understand, we will use a radius of 1, meaning the height (y) is also 1. The point will be (0, 1). tan 90 = y/x = 1/0 tan 90 = undefined tan 90 is undefined because you can't divide 1 by nothing. This is like saying: 1 divided by 0 is ? → ? multiplied by 0 is 1 Nothing multiplied by 0 will give an answer of 1, so the answer is undefined. Answer 3 tan 90 = Not defined Question 4 What is the derivative of y = arcsin (x)? Answer 1 This identity can be proven easily by applying sin to both sides of the original equation: y = arcsin x sin y = sin (arcsin x) sin y = x We continue by using implicit differentiation, keeping in mind to use the chain rule on sin y cos y * dy/dx = 1 Solve for dy/dx dy/dx = 1/cos y Now, substitution with our original equation yields dy/dx in terms of x dy/dx = 1/cos (arcsin x) At first this might not look all that great, but it can be simplified if one recalls the identity sin (arccos x) = cos (arcsin x) = √(1-x²) dy/dx = 1/√(1-x²) This is a good definition to memorize, along with d/dx[arccosx] and d/dx[arctan x] since they appear quite frequently in differentiation problems. Answer 2 There is another way to do it. We have a formula for calculating the derivatives of inverse functions: (f⁻¹)'(x) = 1 / f'(f⁻¹(x)) Here f⁻¹(x) = arcsin x -> f(x) = sin x and f'(x) = cos x 1 So (arcsin x)' = 1 / cos (arcsin x) 1 / cos (arcsin x) = 1 / √(1-x²) Using the arguments given in the other answers, we know that 1 / √(1-x²) Question 5 What is π/6 radians in degrees? Answer 1 A radian is the angle subtended such that the arc formed is the same length as the radius. There are 2π radians in a circle, or 360 degrees. Therefore, π is equal to 180 degrees. 180° / 6 Answer 2 You can convert between radians and degrees using the formula x = (θ * (180/π)), assuming x degrees represents the angle that you want to find. Explanation: x = (π/6) * (180/π) x = 180π/6π x = 180/6 x = 30° So, π/6 measures 30° Answer 3 1rad = 180/π degree ⇒ π/6 rad = π/6 * (180/π) = 30°