12Chapter 1•Right Triangle Trigonometry§1.2Example1.10radicallow330◦ABCDEFradicalBig32radicalBig322145◦Find the sine, cosine, and tangent of 75◦.Solution:Since 75◦=45◦+30◦, place a 30−60−90 right triangle△ADBwith legs of lengthradicallow3 and 1 on top of the hypotenuseof a 45−45−90 right triangle△ABCwhose hypotenuse haslengthradicallow3, as in the figure on the right. From Figure 1.2.2(a) weknow that the length of each leg of△ABCis the length of thehypotenuse divided byradicallow2. SoAC=BC=radicallow3radicallow2=radicalBig32. DrawDEperpendicular toAC, so that△ADEis a right triangle. Since∠BAC=45◦and∠DAB=30◦, we see that∠DAE=75◦sinceit is the sum of those two angles. Thus, we need to find the sine,cosine, and tangent of∠DAE.Notice that∠ADE=15◦, since it is the complement of∠DAE.And∠ADB=60◦, since it is the complement of∠DAB. DrawBFperpendicular toDE, so that△DFBis a right triangle.
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