The Unit Circle Part II (With Trig!!) MSpencer

Multiples of 90°, 0°, 0 360°, 2  180°,  90°, 270°,

The Quadrants (with Angles) 0°, 0 360°, 2  180°,  90°, 270°, Q I 0° <  < 90° 0 <  < QII 90° <  < 180° <  <  QIII 180° <  < 270°  <  < QIV 270° <  < 360° <  < 2 

The Unit Circle r = 1 Remember it is called a unit circle because the radius is one unit. So let’s add in ordered pairs to the unit circle.

Multiples of 90°, 0°, 0 180°,  90°, 270°, (1, 0) (0, 1) (  1, 0) (0,  1) r = 1

45°, 45° Notice that 45° or forms one of the two special right triangles from geometry.

45°, 45° Let’s review this triangle from geometry. Opposite the congruent, 45° angles are congruent sides. These sides are the legs of the right triangle. So the triangle is an isosceles right triangle.

45°, 45° Let’s call the two congruent legs s. s s The hypotenuse is the length of either leg, s, times ; thus, s.

45°, 45° Lastly, now remember that the hypotenuse is the radius of the unit circle, which means it must equal one. Solve for s. s s

45°, 45° 1 The distance across the bottom side of the triangle represents the x- coordinate while the right, vertical side represent y.

Signs and Quadrants 0°, 0 180°,  90°, 270°, Q I (+, +) The signs of each ordered pair follow the signs of x and y for each quadrant. Q II ( , +) Q III ( ,  ) Q IV (+,  )

Multiples of 45°, 135°, 315°, 45°,225°, 45°

60°, Notice that 60° or forms the other special right triangle from geometry. 60° 30°

60°, Let’s review this triangle from geometry. Call the the smallest side opposite 30° s. 60° 30° The hypotenuse is twice the smallest side, or 2s. The medium side opposite 60° is times the smallest side, or. s 2s2s

60° 30° s 2s = 1 60°, The hypotenuse is the radius of the unit circle, which means it must equal one. Solve for s. The medium side opposite 60° is

60°, Notice that since the triangle is taller than it is wide, that the y-coordinate is larger than the x- coordinate. y x

Multiples of 60°, 120°, 300°, 60°,240°,

30°, Notice this is the same special right triangle as for 60° except the x and y coordinates are switched. y x 60° 30°

Multiples of 30°, 150°, 330°, 30°,210°, 60° 30°

Ordered Pairs and Trig From geometry, recall SOHCAHTOA, which defines sine, cosine, and tangent. sine (Sin) = cosine(Cos) = tangent (Tan) =

30°, 60° 30° Ordered Pairs and Trig Cos 30° = cos 30° = Notice that the cosine of the angle is simply the x-coordinate!

30°, 60° 30° Ordered Pairs and Trig Sin 30° = sin 30° = Notice that the sine of the angle is simply the y-coordinate!

 And this is true for ANY angle, often called . cos  = x sin  = y Ordered Pairs: Cosine & Sine  (x, y) (cos , sin  )

Signs for Cosine and Sine 0°, 0 180°,  90°, 270°, Q I (+, +) The “signs” of cosine and “sine” follow the signs of x and y in each quadrant. Q II ( , +) Q III ( ,  ) Q IV (+,  ) So in QII, for instance, cosine is negative while sine is positive.

The Whole Unit Circle Together (Grouped) 0°, 0 (1, 0) 90°, (0, 1) 180°,  (  1, 0) 270°, (0,  1) 45°, 135°, 225°, 315°, 60°, 120°, 240°, 300°, 30°, 150°, 210°, 330°,

The Whole Unit Circle Together (In Ascending Order) 0°, 0 (1, 0) 90°, (0, 1) 180°,  (  1, 0) 270°, (0,  1) 45°, 135°, 225°, 315°, 60°, 120°, 240°, 300°, 30°, 150°, 210°, 330°,