Trigonometric functions and circular measure - the Australian ...

{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