Find the exact value of each trigonometric ratio: A. sin30 B. cos240 C. tan150

Question

Confused with special triangles... do these values even involve special triangles? 

Answer 1

Another method is to use the trigonometric identities. 

 

 

Part B)

 

We can break up the angle as a sum or difference of angles that we can easily find sine and cosine of.   Then apply the addition/subtraction angle identity for cosine.

 

cos(270 - 30) =

 

cos(270)cos(30) + sin(270)sin(30) =

 

  0 * √(3) / 2     +   (-1) * 1/2 =

 

                 -1/2

 

 

Part C)

 

Use the same method from part B combine with the identity  tanθ = sinθ / cosθ.

 

tan(150) =

 

tan(180 - 30) =

 

 

 

sin(180 - 30)

_________________   =

 

cos(180 - 30)

 

 

 

 

sin(180)cos(30) - cos(180)sin(30)

___________________________________  =

 

cos(180)cos(30) + sin(180)sin(30)

 

 

 

 

0 * √(3)/2   -    (-1) * (1/2)

_______________________________  =

 

-1 * √(3)/2   +   0  *  (1/2)

 

 

 

 

(1 / 2) / (-√(3) / 2) =

 

(1 / 2) * (-2 / √3) =

 

-1 / √3

 

 

Rationalize the denominator.

 

-√(3) / 3 

Answer 2

they are all on the unit circle

A. sin 30=1/2   (30º=pi/6)

 

B. cos 240=-√3/2   (240º=4pi/3)

 

C. tan 150= ?   (tan 150 is in the 2nd quadrant where cos is negative and sin is positive; tan=sin/cos so tan is negative)

    tan 150=(1/2)/(-√3/2)

    tan 150=(1/2)(-2/√3)

    tan 150=(-1/√3)=(-√3/√3√3)=-√3/3   (150º=5pi/6)