How do you figure out trigonometric functions of special angles without a calculator?

Question

for example this problem sin pi/4

Answer 1

You can memorize the values or you can draw a unit circle and recognize that pi/4 is 45 degrees. Next you remember that sine and cosine give you the y and x values at that angle on the unit circle

So for pi/4, since

cos(pi/4) = x

and

sin(pi/4) = y

and at 45 degrees

x = y

and you know that for any point on unit the circle

x^2 + y^2 = 1

then you know that

x^2 = y^2 = 1/2

and

x = y = sqrt(2)/2


You can do the same sort of thing for pi/6 and pi/3.

Answer 2

needless to say that's a risk. For the 1st one all human beings comprehend that the sin of the mindset could be 5/6 or that the triangle is created from components 5 and sqrt(11) with a hypotenuse of 6 leaving a cot of the mindset as sqrt(11)/5. For the 2nd one the triangle could be created from components 6 and sqrt(13) with a hypotenuse of 7 leaving a tan of the mindset of sqrt(13)/6. those strategies come from the definition of the sine, cosine, and tangent ratios themselves. as quickly as you're given a topic count like this think of of the arc applications protecting "the sine of the mindset is" or "the cosine of the mindset is" and then use Pythagorean theorem to fill indoors the blanks.

Answer 3

Draw triangles and ask Chief SOH-CAH-TOA.
sin = opposite / hypotenuse
cos = adjacent / hopotenuse
tan = opposite / adjacent

You need to know how many radians are in a complete circle. Then you can figure out what portion of a circle is represented by pi/4.

Answer 4

: pi/6 : pi/4 : pi/3

Sin : 1/2 : 1/sqrt(2) : sqrt(3)/2

Cos : sqrt(3)/2 : 1/sqrt(2) : 1/2

Tan : 1/sqrt(3) : 1 : Sqrt(3)