Krishna
0

Step 1:  Know about the unit circle

      NOTE: Any point on the unit circle will be a distance of one unit from the center

                  this is the definition of the unit circle. 

  •   The terminal side of an angle \theta in standard position intersects the unit circle at  (\cos \theta, \sin \theta).
  •   Another thing you can see from the unit circle is that the values of sine and cosine will never be more than 1 or less than –1.


Step 2: Confirm that the point they have given is a point on the unit circle or not

            NOTE: To confirm, apply the Pythagoras theorem to find the length of the radius of the right triangle formed by dropping a perpendicular from the x-axis down to the point.   

              Pythagoras theorem: hypotenuse^2 = side^2 + side^2

                

                EXAMPLE: Given sides x = \frac{77}{85}, y = \frac{36}{85}

                                  

                                         (\frac{77}{85})^2 + (\frac{36}{85})^2 = radius^2


                                                \frac{(77)^2 + (36)^2}{(85)^2} = radius^2


                                                \frac{5,929 + 1,296}{7,225} = radius^2

                                                         \frac{7,225}{7,225} = radius^2


                                                    Radius = \sqrt{1}


                                                      Radius = 1


                 The Pythagorean Theorem given a value for the radius = 1,  

                  So, I confirmed that the point is on the unit circle.

        

Step 2: Find the trigonometric ratio values by using the unit circle

              NOTE: The terminal side of an angle \theta in standard position intersects the

              unit circle at  (\cos \theta, \sin \theta).


              The given point (\frac{77}{85}, \frac{36}{85})


              Therefore,   \cos \theta = \frac{77}{85}  , \sin \theta = \frac{36}{85}