Brief Summary
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
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
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
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
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
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
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
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
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.
