Trigonometric functions and circular measure - the Australian ...

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{10} • Trigonometric functions and circular measureRelated anglesIn the module Further trigonometry (Year 10), we saw that we could relate the sine andcosine of an angle in the second, third or fourth quadrant to that of a related angle inthe first quadrant. The method is very similar to that outlined in the previous section forangles in the second quadrant.We will find the trigonometric ratios for the angle 210 ◦ , which lies in the third quadrant.In this quadrant, the sine and cosine ratios are negative and the tangent ratio is positive.To find the sine and cosine of 210 ◦ , we locate the corresponding point P in the thirdquadrant. The coordinates of P are (cos210 ◦ ,sin210 ◦ ). The angle POQ is 30 ◦ and is calledthe related angle for 210 ◦ .yP(cos 210 º,sin 210 º)Q210º30 º(cos 30 º,sin 30 º)30 ºxO 1So,3cos210 ◦ = −cos30 ◦ = −2Henceand sin210 ◦ = −sin30 ◦ = − 1 2 .tan210 ◦ = tan30 ◦ = 1 3.In general, if θ lies in the third quadrant, then the acute angle θ−180 ◦ is called the relatedangle for θ.Exercise 2a Use the method illustrated above to find the trigonometric ratios of 330 ◦ .bWrite down the related angle for θ, if θ lies in the fourth quadrant.The basic principle for finding the related angle for a given angle θ is to subtract 180 ◦from θ or to subtract θ from 180 ◦ or 360 ◦ , in order to obtain an acute angle. In the case
A guide for teachers – Years 11 and 12 • {11}when the related angle is one of the special angles 30 ◦ , 45 ◦ or 60 ◦ , we can simply writedown the exact values for the trigonometric ratios.In summary, to find the trigonometric ratio of an angle between 0 ◦ and 360 ◦ :• find the related angle• obtain the sign of the ratio by noting the quadrant• evaluate the trigonometric ratio of the related angle and attach the appropriate sign.ExampleUse the related angle to find the exact value of1 sin120 ◦ 2 cos150 ◦ 3 tan300 ◦ 4 cos240 ◦ .Solution1 The angle 120 ◦ is in the second quadrant, and its related angle is 60 ◦ .3So sin120 ◦ = sin60 ◦ =2 . y1120 º60 ºO1x32 cos150 ◦ = −cos30 ◦ = −2y130 ºO150 º1x
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