TLDR;
Alright, so this video is all about inverse trigonometric functions, innit? We're talking basics, how to find domain and range, some tricky problems, and proving those important properties. Key takeaways include:
- Understanding the principal value range is super important.
- Knowing how to convert between different inverse trig functions is a must.
- Domain and range restrictions can be used to solve problems.
Basics, Domain, Range and Graphs [0:00]
So, a regular trigonometric function takes an angle as input and spits out a value. Inverse trig functions do the opposite, yeah? They take a value as input and give you an angle as the output. Sine inverse x, cos inverse x, all that jazz – they're just angles, okay? But here's the catch: for a function to be invertible, it needs to be one-to-one and onto. That's why we restrict the domain of trig functions to define their inverses properly.
Extreme range related 3 SE [19:42]
The principal value range is the specific range we use to define the inverse trig functions. For sine inverse, it's -π/2 to π/2; for cos inverse, it's 0 to π. Remember these ranges, boss, they're crucial! Now, let's look at a problem: If sine inverse x + sine inverse y + sine inverse z = 3π/2, find the value of a complicated expression. Since the maximum value of sine inverse is π/2, each term must be π/2. This means x = y = z = 1, and you can easily find the answer.
Domain related 5 SE [25:38]
Next up, domain-related problems. Consider f(x) = sine inverse (log base 2 of x/2). To find the domain, we know that -1 ≤ log base 2 of (x/2) ≤ 1. Solving this inequality gives us 1 ≤ x ≤ 4. Sometimes, just knowing the domain can help you solve a problem quickly.
Domain and Range Matching based question [34:53]
Alright, check this out. We've got two sets, E1 and E2, and two functions, f(x) = log(x/(x-1)) and g(x) = sine inverse(log(x/(x-1))). First, find the domain of g(x). Since it involves sine inverse, -1 ≤ log(x/(x-1)) ≤ 1. Solve this inequality to find the domain of g(x), which is E2. Then, find the domain of f(x), which is E1. You'll see that E2 is a subset of E1.
Conversion of inverse trigonometric function [40:17]
Before proving properties, let's see how to convert one inverse trig function to another. If you have sine inverse x, take it as theta. Then, x = sine theta. Use a right-angled triangle or trigonometric identities to find cos theta, tan theta, etc., and convert to cos inverse, tan inverse, and so on.
Proving properties & identities [44:37]
Time for some properties! Sine inverse (-x) = -sine inverse x, cos inverse (-x) = π - cos inverse x, and so on. To prove these, take the inverse trig function as theta, manipulate the equation, and use trig identities. Also, sine inverse x = cosec inverse (1/x), cos inverse x = sec inverse (1/x), and tan inverse x = cot inverse (1/x) (only when x > 0).
Self-adjusting property 1 SE1 [57:20]
If f and f inverse are inverse functions, then f(f inverse(x)) = f inverse(f(x)) = x. This means sine(sine inverse x) = x, cos(cos inverse x) = x, and so on, provided x lies in the domain of the inverse trig function.
Self-adjusting property 1 SE2 [1:00:32]
Let's solve a matching type problem. Simplify a complicated expression involving cos(tan inverse y), sine(tan inverse y), cot(sine inverse y), and tan(sine inverse y). Use the same techniques: take tan inverse y or sine inverse y as theta, draw a right-angled triangle, find the values of cos theta, sine theta, etc., and simplify the expression. You'll find that the expression simplifies to 1.