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4.4 Day 1 Trigonometric Functions of Any Angle – Use the definitions of trigonometric functions of any angle – Use the signs of the trigonometric functions Pg. 499 #2-22 (even) At any point on the circle, we can connect a vertical line to the x-axis and create a triangle. Horizontal side = x, vertical side=y, and hypotenuse=r. x and y may be positive or negative (depending on their direction) The radius, r, is always a positive value. For any point (x,y) found on the circle, sin Θ = y cos Θ = x tan( ) sin( ) y cos( ) x csc ) 1 1 = Reciprocal of Sine sin( ) y sec( ) 1 1 = Reciprocal of Cosine cos( ) x cot( ) cos( ) x sin( ) y Given a point (-3,-4), find the 6 trig. functions associated with the angle formed by the ray containing this point. x=-3, y=-4, and so r = (3) 2 (4) 2 25 5 Since, sin A = y/r, cos A = x/r, tan A = y/x csc A = r/y, sec A = r/x, cot A =x/y Then, sin A = -4/5, cos A = -3/5, tan A = 4/3 csc A = -5/4, sec A = -5/3, cot A = ¾ 1. Let P = (1, -3) be a point on the terminal side of Θ. trigonometric functions of Θ. Find each of the six Evaluate, if possible, the cosine function at the following four quadrant angles: 2. 0o 3. 90o 4. 180o 5. 270o Evaluate, if possible, the cosecant function at the following four quadrant angles: 6. 0o 7. 90o 8. 180o 9. 270o Here’s a helpful anagram to help you remember the signs (positive or negative) of trigonometric functions: All All functions Snowflakes Sine and cosecant Taste Tangent and cotangent Cold Cosine and secant are (+) in Quadrant I are (+) in Quadrant II are (+) in Quadrant III are (+) in Quadrant IV Another way to think of the sign of a function is to remember the variable it is defined by. Since the radius is always positive, only the signs of x and y influence the sign of the function. In any given quadrant: cosine and secant have the same sign as x sine and cosecant have the same sign as y tangent and cotangent have the same sign as the ratio of x and y Determine the quadrant in which angle Θ lies. 10. sin Θ < 0 and cos Θ > 0 11. sin Θ < 0 and tan Θ > 0 12. sec Θ > 0 and cot Θ < 0 13. tan Θ < 0 and csc Θ < 0