Thank you sir

Sir . how to find principal value
Cos inverse of (-1/3) in radian

The principal values for cos x are angles of x between 0 and 180 (pi), or the first two quadrants.

cosinv (-1/3) must be in the second quadrant as cos in the second quadrant is negative.

So, this is as good as finding the value of x in the second quadrant for which cos x = -1/3.

Basic angle
= cosinv (1/3)
= 1.230959417 rad

Angle required
= second quad’s angle
= pi - basic
= 1.910633236 rad
~ 1.91

Sorry sir how to find principal value for sin and tan. Please explain sir i don't know about principal value

sin⁴ 2x + cos⁴ 2x

= (sin² 2x)² + 2(sin² 2x)(cos² 2x) + (cos² 2x)² - 2(sin² 2x)(cos² 2x)

= (sin² 2x + cos² 2x)² - 2(sin 2x cos 2x)²

= 1² - 2(½sin 4x)²

= 1 - ½ sin² 4x

= ¾ + ¼(1 - 2sin² 4x)

= ¾ + ¼cos 8x


d/dx (sin⁴ 2x + cos⁴ 2x)

= d/dx (¾ + ¼ cos 8x)

= ¼ (-sin 8x)(8)

= -2sin 8x

k = -2, p = 8

Principal value is not taught in some schools for some reason. My two Sec 4 students did not even learn this in their schools.

For sine, the principal values lie within -90 and 90 (inclusive of both), while for tan, the principal values lie within -90 and 90 (excluding both).

It’s hard for me to explain in words, but it’s basically the universally accepted minimal values of x which gives rise to half a cycle of the graph containing the minimum and maximum y values of a graph, including the tangent case where this stretches to infinity.

https://www.math-only-math.com/principal-values-of-inverse-trigonometric-functions.html

Sadly enough not all schools teach this.

And one of my Sec 4 students who yesterday had a difficult time understanding the picking out the terms in x^3 for a binomial expansion of (...) (...)^9 as I explained it to him. He preferred expanding fully.

Well, sometimes there are things in the syllabus which not all students actually learn or absorb well.

Picking out terms selectively is just a matter of visualisation. It involves picking out the terms that when multiplied together, give a product that is part of the required terms.

The reasons why they prefer expanding fully is to lay out all the terms and then select from there. Perhaps it works better for them in the sense that they don't miss out any terms, which could happen in the former

Principal values are probably not taught as the inverse functions themselves are not taught at O level yet. IP stream students may learn it early though.

They're just using arcsin/arccos/arctan to get their required angles , based on ASTC.

And not many would know sin-¹ x is also called arcsin x. It's called arcsin because of the arc subtends an angle at the centre whose sine is x.

eg. sin-¹ 1 = 90° or π/2 rad

The arc that subtends this angle of the unit circle has length of π/2 r, where r is the radius. This corresponds to ¼ of the circumference (which is 2πr)

(i.e Arc length = rθ

Thinking in degrees would be 90°/360° x 2πr

MM, this will make more sense to you learn the concept of functions at higher levels.

For the inverse function to be defined (sin-¹x , cos-¹x, tan-¹x etc), the function itself must be injective (one-one, every value of sinx, cosx, tanx can only have at most 1 value of x corresponding to it).

Now we know sinx cosx and tanx can go on and on, so we need to restrict the domain (the set of values of x where we plot/define this graph).

If not, there will be many values of x corresponding to each value of sinx , cosx, tanx and we can't plot the inverse.


So this domain is restricted to those values which we call the principal values