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Aim: How do we find the values of trig functions of quadrantal angles? Do Now: Given a unit circle and triangle AOB: M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com Find the values of a) sin 1 b) cos c) tan HW: p.380 # 20,22,24,26,28,30,32 p.395 # 22,42,56(a,b,c) A (0.8, 0.6) -1 O -1 1 Remember: in unit circle y tan x counterclockwise, x y sin y, cos x r r If we rotate the terminal side OA the value of y will keep increasing. Finally, the terminal side will coincide with the y-axis and at this moment x = 0 and y = 1. The 90 Therefore, we say that Sin90 1 If we keep rotating the terminal side, it will coincide with the negative x-axis at (-1,0). Therefore, we know that Sin180 0 Sin270 1 and Sin360 / 0 0 . . In the same fashion, we can get cos 90 0, cos180 1 , cos 270 0 and cos 360 / 0 1 0 tan 0 0 1 1 tan 90 (undefined) 0 0 tan 180 0, 1 1 tan 270 (undefined) 0 Quadrantal Angles Try This 90 • Find the six trigonometric for 270values . 1. Draw the angle. Step Step 2. Find the ordered pair from the unit circle.. Step 3. Apply the definitionswe learned from the reference angle to find the trigonometric values. ( 0 ,1 ) 180 ( 1 , 0 ) sin 270 y 1 r 1 1 x 0 cos 270 0 r 1 y tan 270 1 undefined x 0 (1, 0 ) 360 1 270 ( 0, 1 ) In the unit circle, the radius = 1 that means the hypotenuse of any right triangle is also 1 1 opp sin opp. 1 adj cos adj. 1 opp sin tan adj cos adj cos cot opp sin They are called quotient identity How do we use quotient identity to solve problems? 2 2 1 P 3 , 3 on A point and cot M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com y 1 x θ -1 P 1 -1 the unit circle, find tan 1 sin tan 3 cos 2 2 1 2 3 4 2 2 1 4 4 2 cot 2 2 tan 2 2 2 A circle with center at (0, 0) and radius 1 is called a unit circle. (0,1) (-1,0) (1,0) (0,-1) The points on the circle must satisfy x2 + y2 = 1 For angles that are multiples of 30 they only have one of 2 possible values for x and y if they don’t land on an axis. If the x is further you know that one is the square root of 3 over 2, if it is not as far over as the y is up or down you know it is ½. 1 3 , 2 2 120° 3 1 , 2 2 1,0 0,1 90° 1 3 , 2 2 60° 150° 30° 3 1 , 2 2 0° 1,0 180° 210° 3 1 2 , 2 330° 240° 1 3 , 2 2 270° 300° 0,1 3 1 , 2 2 1 3 , 2 2 1. sin 630 2. cos( 630) 3. tan( 900) 4. cos1260 5. tan 720 6. (sin 0)(tan 30) cos 0 7. sin 45 cos 45 sin 90 cos 90 8. tan( 120) sin 30 cos 30 tan 0 9. cos 405 sin( 45) tan 135 cos( 30)