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Trigonometric Functions Cosecant is reciprocal of sine. Secant is reciprocal of cosine. Cotangent is reciprocal of tangent. Suppose the terminal side of a rotation angle (K) passes through the point (–3, 4). Draw angle K in standard position. Find the distance from (–3, 4) to the origin using r x 2 y 2 . r 5 Evaluate the six trig functions for this angle. sin K 4 5 cos K 3 5 tan K 4 3 csc K 5 4 sec K 5 3 cot K 3 4 Suppose angle K measures approximately ―233.13°. Do the calculator values for sine, cosine, and tangent of this angle match the answers you previously found for angle K? sin K 4 5 sin 233.13 0.8 cos K 3 5 cos 233.13 0.6 tan K 4 3 tan 233.13 1.333 A reference triangle can be formed by drawing a perpendicular segment from a point on the angle’s terminal side to the x-axis. This perpendicular segment (a y-value) can act as the “opposite” side, and its distance from the distance from origin (an x-value) serves as the “adjacent” side. Here, however, the opposite and adjacent legs can be negative. x = “adjacent” leg y = “opposite” leg These values, because they are coordinates, can be positive or negative (or even zero). r = “hypotenuse” The r-value, however, because it is a distance, is defined to always be positive. Angle: θ, Point: (12, –5) A) Sketch the angle in standard position, and indicate the Quadrant where the terminal side lies. B) Find the value of r using r x2 y 2 . r 13 Quadrant IV C) Evaluate the six trig functions for this angle. sin 513 cos 12 13 tan 512 csc 13 5 sec 1312 cot 12 5 Angle: α, Point: (–1, 2) A) Sketch the angle in standard position, and indicate the Quadrant where the terminal side lies. B) Find the value of r using r x2 y 2 . r 5 Quadrant II C) Evaluate the six trig functions for this angle. 2 5 sin 5 5 csc 2 5 cos 5 sec 5 tan 2 1 cot 2 Angle: Z, Point: (–3, –3) A) Sketch the angle in standard position, and indicate the Quadrant where the terminal side lies. B) Find the value of r using r x2 y 2 . r 3 2 Quadrant III C) Evaluate the six trig functions for this angle. 2 sin 2 csc 2 2 cos 2 sec 2 tan 1 cot 1 Angle: P, Point: (24, 7) A) Sketch the angle in standard position, and indicate the Quadrant where the terminal side lies. B) Find the value of r using r x2 y 2 . r 25 Quadrant I C) Evaluate the six trig functions for this angle. sin 7 25 25 csc 7 24 cos 25 25 sec 24 7 tan 24 24 cot 7 Based on the answers to these problems, tell in which Quadrant the following are always true. • Only sine (and cosecant) values are positive Quadrant II • Only cosine (and secant) values are positive Quadrant IV • Only tangent (and cotangent) values are positive Quadrant III • All trig ratio values are positive Quadrant I Label the Quadrants in which the six trig functions are positive. sin, csc sin, cos, tan, cot,sec, csc tan, cot cos,sec 3 sin , Quad II 5 3 A) Sketch the angle in standard position B) Find the missing value 5 -4 x r 2 y2 . x 4 x’s are negative in Quad II C) Evaluate the other trig functions for this angle. sin 3 5 5 csc 3 4 cos 5 5 sec 4 3 tan 4 4 cot 3