need help with part a only, dont understand what is principal value and why is there a - 1 at sinx and cosx

That -1 stands for the inverse function.

eg. A function f(x) has the inverse function f-¹(x).

It is not to be confused with reciprocal of a term eg. x-¹ = 1/x

sin-¹x and 1/sinx are not the same thing.

Your teacher should have already gone through this.

You can think of it as the reverse of the functions.

Eg. If sin 90° (or sin π/2 in radians) = 1, then the inverse is :

sin-¹ 1 = 90° (or π/2)

sorry im still confused ;-;

As for the principal value :

For functions of sin x and cos x, they are periodical. You can have multiple input values of x giving the same result/output.

Eg. sin 0 = 0 , but sin π , sin 2π, sin 3π, sin 4π, etc all also = 0.


Generally these functions repeat every 360° (or every 2π in radians,hence the name 'periodic'.


In a typical function, every 1 input corresponds to 1 output only.

i.e 1 value of x gives you 1 value of y, not 2, not 3 , not 10 values.

Example y = x².

When x = 3, y = 9 and not any other value.

However, outputs may not necessarily be unique.

Eg. when x = -3, y = 9. So y = 9 is not unique to x = 3 only. But it is still a case of 1 input leading to 1 output.



So for the inverse function of sin x and cos x to exist, each value of x can only correspond to 1 value of sin-¹ x and cos-¹ x


But the problem here is, for sin-¹x, an input like x = 0 would mean the output sin-¹ 0 could lead back to 0, π, 2π, 3π, etc.

So it can't be a function as 1 input of x would lead to multiple values of sin-¹ x.


How to overcome this ? Use principal value.

We only define 1 output for each input by setting this principal value. This means each input of x only corresponds to 1 output of sin-¹x (the term principal means the main)


Since 1 ≤ sin x ≤ 1 and -1 ≤ cos x ≤ 1, the principal value is set for the inverse functions sin-¹ x and cos-¹ x based on -1 ≤ x ≤ 1.


-π/2 ≤ sin-¹ x ≤ π/2

(Recall the sin x graph. It ranges from -1 at x = -π/2 to 1 at x = π/2. So inverse ('reversing') of 1 and -1 gives you back those values)

Likewise,


0 ≤ cos-¹ x ≤ π

(Recall the cos x graph. It ranges from 1 at x = 0 to -1 at x = π)