Thursday, June 16, 2011

5.1 – Inverse Trigonometric Functions

This chapter will be of less words, but more formulas. What you need to do in this chapter is:
1. memorize the useful graphs, identities and formulas.
2. spend your time trying to derive all the identities.

With this 2 points done, you are sure to score for this chapter. STPM questions will be about proving them, sketching graphs, or differentiating and integrating them (which will be covered in the next chapter).


You have learnt about trigonometric functions throughout your secondary school years. Now, we let sin y = x. An inverse trigonometric function inverses the trigonometric function, and is denoted as y = sin-1 x.

Note that there is a difference between sin-1 x and (sin x)-1. This is only one of the 6 inverse trigonometric functions, the rest of them are cos-1 x, tan-1 x, sec-1 x, csc-1 x, and cot-1 x.

Following are the graphs of the 6 inverse trigonometric functions:
Inverse-trigonometric-functions-picture

The domain and the range of the functions are as follows:
image


Now that you the details about these 3 inverse trigonometric functions, it’ll be formulas and identities. Try to remember as many as you can. In fact, make sure you know how to derive every single one of them.

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Prove the first one by letting x = cos y, the rest follows.

Inverse-Forward Identities
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Forward-Inverse Identities
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Proving this one is not hard too. Make x = cos y, and make use of the identity cos2 x + sin2 x = 1. The rest follows too. Just that probably the tan(cos-1 x) one will be harder. Give it a try.

Inverse Sum Identities
image
Prove the first one by letting x = cos (π/2 – y) = sin y. Try figuring out the rest yourself.

sin-1 (-x) = –sin-1 x
csc-1 (-x) = –csc-1 x
cos-1 (-x) = π – cos-1 x
sec-1 (-x) = π – sec-1 x
tan-1 (-x) = –tan-1 x
cot-1 (-x) = –cot-1 x
This one is proven by letting sin y = x, and sin –y = –x. The rest follows.

image 
I don’t think this one will come out in exams. However, the proof requires you to learn the inverse hyperbolic in the next section first.

image
I’ll leave this proof to you to try.

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This is one is the hardest to prove. Try proving using the formula
image
You probably don’t even know that this formula exist.


Happy deriving.

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