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**Aim: What good is the Unit Circle and how does it help us**to understand the Trigonometric Functions? Do Now: A circle has a radius of 3 cm. Find the length of an arc cut off by a central angle of 2700.**Q II**Quadrant I terminal side 90 < < 180 0 < < 90 terminal side initial side t.s. Q III Q IV t.s 180 < < 270 270 < < 360 • An angle on the coordinate plane is in standard position when its vertex is at the origin and its initial side coincides with the nonnegative ray of the x-axis. Angles in Standard Position y x • An angle formed by a counterclockwise rotation • has a positive measure. • Angles whose terminal side lies on one of the axes is • a quadrantal angle. i.e. 900, 1800, 2700, 3600, 4500 etc.**Q II**Quadrant I 90 < < 180 0 < < 90 initial side - t.s. Q III Q IV 180 < < 270 270 < < 360 Co-terminal and Negative Angles y 3000 = x 600 • An angle formed by a clockwise rotation has a • negative measure • Angles in standard position having the same • terminal side are co-terminal angles.**Q II**Quadrant I 90 < < 180 0 < < 90 Q III Q IV 180 < < 270 270 < < 360 • Angles whose terminal side rotates more than one • revolution form angles with measures greater • than 3600. Angles Greater than 3600 y 4850 1250 x • To find angles co-terminal with an another angle • add or subtract 3600. 1250 and 4850 are co-terminal**Model Problems**• Find the measure of an angle between 00 and 3600 co-terminal with • 3850 b) 5750 c) -4050 • In which quadrant or on which axis, does the terminal side of each angle lie? • a) 1500b) 5400 c) -600 215o 315o 25o x-axis QIV QII**hypotenuse**side opp. cos side adj. Unit Circle y 1 radius = 1 center at (0,0) cos , sin (x,y) x -1 1 -1**Aim: What good is the Unit Circle and how does it help us**to understand the Trigonometric Functions? Do Now: Find the measure of an angle between 00 and 3600 co-terminal with an angle whose measure is -1250.**3**Hypotenuse = 2 shorter leg Longer leg = shorter leg Value of Sine & Cosine: Quadrant I y 1 radius = 1 center at (0,0) cos 600, sin 600 (x,y) 600 x -1 1 What is the value of coordinates (x,y)? 300-600-900 triangle Sine and Cosine values for angles in Quadrant I are positive.**(x,y)**1200 side adj. 3 Hypotenuse = 2 shorter leg Longer leg = shorter leg Sine values for angles in Quadrant II are positive. Value of Sine & Cosine: Quadrant II y 1 cos 1200, sin 1200 What is the value of coordinates (x,y)? 1 60º is the reference angle (180º-120º) 600 x -1 1 directed distance A reference angle for any angle in standard position is an acute angle formed by the terminal side of the given angle and the x-axis. What is the cosine/sine of a 1200 angle? 300-600-900 triangle Cosine values for angles in Quadrant II are negative.**side opp.**side adj. Value of Sine & Cosine: Quadrant III y 1 What is the value of coordinates (x,y)? What is the cosine/sine of a 2400 angle? 2400 directed distance 60º is the reference angle (240º-180º) x -1 600 1 directed dist. 1 (x,y) cos 2400, sin 2400 Sine and Cosine values for angles in Quadrant III are both negative.**side opp.**(x,y) Sine values for angles in Quadrant IV are negative. Value of Sine & Cosine: Quadrant IV y 1 What is the value of coordinates (x,y)? What is the cosine/sine of a 3000 angle? 60º is the reference angle (360º-300º) 3000 x -1 600 1 directed dist. 1 cos 3000, sin 3000 Cosine values for angles in Quadrant IV are positive.**Periodic**Unit Circle – 8 Equal Arcs Negative Angles Identities**y**Quadrant II Quadrant I x Quadrant III Quadrant IV Value of Sine & Cosine in Coordinate Plane cos is + sin is + cos is – sin is + cos is + sin is – cos is – sin is – for any angle in standard position is an acute angle formed by the terminal side of the given angle and the x-axis. The reference angle:**Model Problems**• Fill in the table • Quad. Ref. sin cos • 2360 • 870 • -1600 • -36 • 13320 • -3960**y**1 x -1 1 -1 Regents Prep On the unit circle shown in the diagram below, sketch an angle, in standard position, whose degree measure is 240 and find the exact value of sin 240o.**Aim: What good is the Unit Circle and how does it help us**to understand the Trigonometric Functions? Do Now: Use the unit circle to find: a. sin 1800 () b. cos 1800****(-1,0) Model Problems Use the unit circle to find: a. sin 1800 () b. cos 1800 (x, y) = (-1, 0) sin 1800 = y = 0 cos 1800 = x = -1 180º - quadrantal angle**sin **sin cos cos = 1 -1 ( 1, tan) ( , )? Tan radius = 1 center at (0,0) cos , sin y 1 (x,y) tan 1 x -1 1**Trigonometric Values**+ – + –**Quadrant I**Q II 90 < < 180 0 < < 90 Q III Q IV 180 < < 270 270 < < 360 Trigonometric Values - A C T S y S Sine is + A All are + x T Tangent is + C Cosine is +**1**y r y x -1 x 1 -1 Reciprocal Functions Negative Angles Identities csc = 1/y sec = 1/x cot = x/y denominators 0 Need to Knows When r = 1 sin = y cos = x tan = y/x**y**(x,y) 1 1 sin = y 2 2 450 x -1 1 cos = x In a 450-450-900 triangle, the length of the hypotenuse is times the length of a leg. -1 (x) length of hypo. = Model Problems Using the unit circle, find cos 450 (/4) sin 450 tan 450 450-450-900 triangle A 450-450-900 triangle is an isosceles right triangle. therefore x = y cos = sin **y**1 1 sin = y 45º x -1 1 cos = x = = -1 Model Problems Using the unit circle, find cos 45º(/4) sin 45º tan 45º (x,y) cos 45º = x tan 45 = 1 sin 45º = y****0º 0 30º /6 45º /4 60º /3 90º /2 sin 0 1 cos 1 0 tan 0 1 UND. Trigonometric Values for Special Angles Why is tan 90º undefined? What is the slope of a line perpendicular to the x-axis? = slope**What is the cos 510º (17/6)?**• cos 30º = • cos 510º= Model Problems What is the tan 135º (3/4)? • 135º is in the 2nd quadrant • 45º is reference angle (180 – 135 = 45) • tan 45º = 1 • tangent is negative in 2nd quadrant • tan 135º= -1 • 510º is in the 2nd quadrant • (510 – 360 = 150) • 30º is reference angle (180 – 150 = 30) • cosine is negative in 2nd quadrant ≈ -.866…**Model Problems**Given: sin 68o = 0.9272 cos 68o = 0.3746 Find cot 112o • -0.3746 B) -2.4751 • C) -0.404 D) 1.0785 reference angle for 112o is 68o; 112o is in QII; tan and cot are negative in QII WHAT ELSE DO WE KNOW?**Model Problems**Express sin 285º as the function of an angle whose measure is less than 45º. What do we know? 285º in IV quadrant the sine of a IV quadrant angle is negative -sin 75º reference angle for 285º is (360 – 285) = 75º > 45º sine and cosine are co-functions complement of 75º is 15º < 45º sin 285º = = -cos 15º -sin 75º**Trig Functions Using Radian Measures**Algebraically: Find: sin (π/3) remember: π/3 radians π/3 60º sin 60º = ≈ .866… Using the calculator: Use the mode key: change setting from degrees to radians then hit: sin 2nd π ÷ ENTER 3 Display: .8660254083**y**1 1 -1 unit circle 1 x -1 Un-unit circle is any angle in standard position with (x, y) any point on the terminal side of and r 1**4**r = 5 3 Model Problem (-3, 4) is a point on the terminal side of . Find the sine, cosine, and tangent of . Q II**-1**r = 2 Model Problem is a point on the terminal side of . Find , the sine, cosine, and tangent of . Q III**Model Problem**Tan = -5/4 and cos > 0, find sin and sec When tangent is negative and cosine is positive angle is found in Q IV.**Model Problem**The terminal side of is in quadrant I and lies on the line y = 6x. Find tan ; find . y = mx + b - slope intercept form of equation m = slope of line y = 6x m = 6 = tan Q I**Model Problem**The terminal side of is in quadrant IV and lies on the line 2x + 5y = 0. Find cos . y = mx + bslope intercept form of equation tan = m = -2/5**y**1 1 45º x -1 1 -1 Templates