I honestly have a problem with most people answering the question. All they did was give you the answer without explaining where the number came up. You're not going to believe me but I literally remembered the entire Unit of Circle when I took Trigonometry.
The rule is very simple. I follow in the format of (cos,sin) with tan being sin/cos. So all I do is look at the side with √3/2 on the x-axis and find out that pi/6 and 11pi/6 has those. Although the calculator will normally assume pi/6. So it all just comes down to memorizing the Unit of Circle to solve these questions without a calculator.
ADDITIONAL INFORMATION: If you use Pythagorean Theorem, you will notice the result is always run. You can draw a triangle based on where you start. Let's go with the one with 30 degrees. If you solve with x^2+y^2 then you get 3/4+1/4=1 and you get that for everything else. Obviously it makes sense because a circle has the same distance from the center.
If you are wondering where cos^2(x)+sin^2(x)=1 came from then think of it like this. X is adjacent, y is opposite, and r is hypotenuse. Think of SOH CAH TOA. Sin=opposite/hypotenuse, Cos=adjacent/hypotenuse, and Tan=opposite/adjacent. Now let's switch it to variables to make it easier. Sin=y/r, Cos=x/y, Tan=y/x
Let's use Pythagorean Theorem now. x^2+y^2=r^2. Divide everything by r^2 and you get x^2/r^2+y^2/r^2=1 which can convert back to Cos^2(x)+Sin^2(x)=1. You can toy around with it and divide everything by Cos^2(x) and you will get 1+Tan^2(x)=Sec^2(x). Or if you choose to divide by Sin^2(x) then you get Cot^2(x)+1=Csc^2(x).
I know this is not related to the question but I just wanted to add something extra to explain where some of these equations come from.
Answers & Comments
Verified answer
In maths, we use specific triangles for these exact values. Please refer to the diagram below.
I honestly have a problem with most people answering the question. All they did was give you the answer without explaining where the number came up. You're not going to believe me but I literally remembered the entire Unit of Circle when I took Trigonometry.
The rule is very simple. I follow in the format of (cos,sin) with tan being sin/cos. So all I do is look at the side with √3/2 on the x-axis and find out that pi/6 and 11pi/6 has those. Although the calculator will normally assume pi/6. So it all just comes down to memorizing the Unit of Circle to solve these questions without a calculator.
ADDITIONAL INFORMATION: If you use Pythagorean Theorem, you will notice the result is always run. You can draw a triangle based on where you start. Let's go with the one with 30 degrees. If you solve with x^2+y^2 then you get 3/4+1/4=1 and you get that for everything else. Obviously it makes sense because a circle has the same distance from the center.
If you are wondering where cos^2(x)+sin^2(x)=1 came from then think of it like this. X is adjacent, y is opposite, and r is hypotenuse. Think of SOH CAH TOA. Sin=opposite/hypotenuse, Cos=adjacent/hypotenuse, and Tan=opposite/adjacent. Now let's switch it to variables to make it easier. Sin=y/r, Cos=x/y, Tan=y/x
Let's use Pythagorean Theorem now. x^2+y^2=r^2. Divide everything by r^2 and you get x^2/r^2+y^2/r^2=1 which can convert back to Cos^2(x)+Sin^2(x)=1. You can toy around with it and divide everything by Cos^2(x) and you will get 1+Tan^2(x)=Sec^2(x). Or if you choose to divide by Sin^2(x) then you get Cot^2(x)+1=Csc^2(x).
I know this is not related to the question but I just wanted to add something extra to explain where some of these equations come from.
Let y=cos^-1[sqr(3)/2] then
cos(y)=sqr(3)/2
=>
y=360n*+/-30*, where n is an integer.
For 0*=<y<=360*,
y=30* or 330*
Sketch a right angled triangle with sides 1 , √3 and 2
This triangle has sides 1 , √3 and 2 and angles 30° , 60° , 90°
From this triangle :-
cos 30 ° = √3 / 2
Arccos[√(3)/2] = 30° answer//
get familiar with the unit circle and common angles
arc cos( x ) = y
is the same as saying
cos(y) = x
∴
arc cos( √3 / 2 ) = y
cos( y ) = √3/2
y = 30° or ⅙π
arc cos( √3 / 2 ) = 30° or ⅙π
———————————
I do not know the answer because I suck at trig...But a good math resource that shows solutions and steps is symbolab.com
cos(π/6) = √3/2 ---> Arc cos(√3/2) = ....
I honestly don't know.