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T3 MATHEMATICS SUPPORT CENTRE Title: Trigonometric Ratios for any Angle Target: On completion of this worksheet you should know the definitions for the sine, cosine and tangent of any angle and be able to write these in terms of an acute angle. Consider a circle of radius r : y Suppose OP has turned through an angle of 2300 P (x,y) y x O N N O x P (x,y) Suppose OP is a rotating arm starting along the x-axis and turning anti-clockwise: OP = r (radius of circle) PN = y (y co-ordinate of point P) ON = x (x co-ordinate of point P) Let the angle turned through be θ 0 ie ∠ PON = θ 0 We define: PN y = OP r ON x cosθ = = OP r PN y tanθ = = ON x sin θ = These definitions are consistent with our previous definitions for right angled triangles. Note: See graph sheet G14 for the graphs of sine, cosine and tangent. Mathematics Support Centre,Coventry University, 2000 The point P has both the x and y coordinates negative so using the definitions we have: y <0 ( r > 0) r x cos 230 0 = < 0 r y tan 230 0 = > 0 x In fact sin 2300 = -0·766. Use your calculator to check that cos 2300 is negative and tan 2300 is positive. Also considering triangle OPN, ∠NOP = 2300 – 1800 = 500 y PN sin 500 = = OP r so sin 2300 = - sin 500 (check this result) Similarly cos 2300 = - cos 500 and tan 2300 = tan 500 sin 230 0 = If the angle turned through by OP is θ then if θ is between: 00 and 900 it is in the first quadrant 900 and 1800 it is in the second quadrant 1800 and 2700 it is in the third quadrant 2700 and 3600 it is in the fourth quadrant 0 rd So 230 is in the 3 quadrant and for any angle in this quadrant the sine and cosine are negative and the tangent is positive. Similarly we find the following: sin θ + + - Quadrant 1st 2nd 3rd 4th cos θ + + tan θ + + - This can be shown in a diagram as follows: Sin only positive All positive Tan only positive Cos only positive Exercise Without using a calculator state which of the following are positive and which are negative: 1. sin 1500 2. cos 3050 3. tan 950 4. sin 2500 (Answers: positive, positive, negative, negative) Examples Write the following in terms of an acute angle: 1. cos 1200 1200 This angle is 600 and since 1200 is in the 2nd quadrant cos 1200 is negative so cos 1200 = - cos 600 2. tan 1900 1900 angle = 1900 - 1800 = 100 1900 is in the 3rd quadrant where tan is positive so tan 1900 = tan 100 This is usually abbreviated as follows: S A T C Examples Without using a calculator state which of the following are positive and which are negative: 0 1. cos 150 This is in the 2nd quadrant where cos is negative so cos 1500 < 0 2. tan 2100 This is in the 3rd quadrant where tan is positive so tan 2100 > 0 3. sin 4000 This is in the 1st quadrant (one complete turn and then 400) so sin 4000 > 0 Mathematics Support Centre,Coventry University, 2000 To find a trig ratio in terms of an acute angle: • Mark the angle on a sketch • Find the acute angle made with the horizontal ie the 00/1800/3600 line • Use the diagram opposite to find whether the answer is positive or negative Exercise Write the following in terms of an acute angle: 1. sin 3000 2. sin 2340 0 3. tan 100 4. cos 1450 5. cos 3900 6. sin 950 0 7. tan 285 8. tan 2500 9. cos 3100 10. sin 4200 0 11. tan 740 12. cos 2750 (Answers: -sin 600, -sin 540, -tan 800, -cos 350, cos 300, sin 850, -tan 750, tan 700, cos 500, sin 600, tan 200, cos 850)