Identities, Forms
The Pythagorean identity of sec^2(x).
What is tan^2(x)+1?
"Now how did you derive that?"
You divide both entire sides of sin^2+cos^2=1 by sin^2 and that simplifies into the answer.
The output of an inverse trig function.
What is the angle which is derived from the trig ratios?
"Correct, hey, between you and me, just curious here but, what is arctan(sin(pi/3))?"
Oh, well, sine at pi/3 is the square root of 3 over 2 and lies at 60 degrees in the first quadrant, so it would be arctan(square root of 3 divided by 2).
The formula for the arc length of the unit circle.
What is S=(radius)(theta)?
"Can you also tell us what the arc length of the moon is if its radius is 1,080 miles?"
Well it's 1,080 times (2pi) which is 2160pi and to undo pi you multiply by pi to get a natural number, 21318.34551 and then divide by pi again to undo that and you get approximately 6785.840132 miles.
This form of Sin(x) reflects over the apexes of the maximum and minimum points of the function.
What is 1 over sin(x)?
"Absolutely!"
This angle is often used to refer to the angle of the line of sight upward to an object in the perspective of an observer below.
What is the angle of elevation?
The Pythagorean Identity of 1+cot^2(theta).
What is csc^2(theta)?
"Correct."
Arcsin(sin(pi)).
What is 0?
The formula for the area of a sector.
What is (1/2)r^2(theta)?
"That is correct. Could you possibly also tell us what the area of the unit circle is?"
Well (1/2)(1)^2(2pi) which is half of 2pi which means the area of the unit circle itself is PI!
This formula is used to find periodicity of trig functions.
What is 2pi/B?
"Tremendous!!! NOW, let's say I have a sin function, like the ones used to bleep out cuss words on television, high pitched, something like 2sin(12x-2pi), can you tell me what in the world is going on there?"
Hm, sounds like the function has an amplitude of 2, so a max at 2 and min at -2, periodicity of 2pi/12 or pi/6, and shifts right 2pi).
Oh, so that's what that noise is?
This is commonly referred to as the opposite side of the angle but in application can be thought of as this.
What is the height?
"Correct, another acceptable answer would be altitude"
If two of my sides are both 1 then my hypotenuse is.
What is square root of 2?
"Correct. What are my angles?"
45, 45, and 90.
That is also correct. But what if one side is 2 and another is 1 then what is my hypotenuse?"
Square root of 3 with angles 30, 60, and 90.
"Correct, you don't win any extra money for side-questions, just a more grandiose sense of accomplishment."
Cos(arccos(pi/2)). extra points to convert it to degrees.
What is pi/2? Or what is (180/pi * pi/2) 90 degrees?
The primary difference between the conterminal angle and the reference angle.
What is bidirectional rotation reference of 360 degrees?
"Absolutely correct! Can you also give me the reference angle for 110 degrees?"
Sure, just minus 180 from 110 and you get 70 degrees, or, the acute angle to the horizontal axis.
"My God, this kid's a natural!!!"
Sin(x)^2, tan(x)^2, cos(x)^2 all share this similarity.
What is positive range?
"Now let me ask you something crazy, what if I asked you to multiply all sin(x)cos(x)tan(x)cot(x)sec(x)csc(x) together, what do you get?
Um, well, all the trig functions of the right triangle multiply with their reciprocal and end up as equaling 1, so it's 1*1*1 =1.
"I would have never thought of that."
Side of a triangle length is called this when given direction and magnitude.
What is a vector?
"You got it."
Sin(x) as written in one of the forms of cosine.
What is cos((pi/2)-x)?
Axis along which the graph of an inverse function reflects describe as an equation.
What is y=x?
"You've got it."
The type of angles used as defining coordinates of the Unit Circle.
What is special angles?
Sin(x)+Cos(x) = this kind of function...
What is a cosine function?
This foundationally trigonometric problem describes the interaction of multiple celestial bodies. Also used by NASA to launch its satellites outside of the solar system.
What is the Three-Body-Problem?
"Spectacular!"
This is an outdated trig identity described as 1-cosine.
What is versine?
"Correct! You've just won 500 cents, how do you feel right right now, the crowd wants to know!?"
Last for this column
The domain of arctan(x).
INFINITY/All REAL NUMBERS
This was inscribed inside of a circle and the sides multiplies by Archimedes in order to prove the existence of pi!
What is a polygon?
"Absolutely, Positively, Correct!"
Sin(x)Cos(x) = this kind of function...
What is a sine function?
Length of distance traveled by a ball thrown from the top of a 100ft building at a 40 degree angle of depression, disregard physical qualia and specifications.
What is hyp=(tan(40)(100))/sin(40)?
"Can you round that to the second decimal place?"
What is 130.54ft?
"Precisely!"