How can you find the exact value of [tex]\tan 75^\circ[/tex]?
Answer (2)
tan \alpha = \frac{sin \alpha }{cos \alpha } \\--------\\\\tan75^0= \frac{\big{sin75^0}}{\big{cos75^0}} \\\\sin75^0=sin(45^0+30^0)=sin45^0\cdot cos30^0+sin30^0\cdot cos45^0=\\\\.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \frac{ \sqrt{2} }{2} \cdot \frac{ \sqrt{3} }{2} + \frac{1}{2} \cdot\frac{ \sqrt{2} }{2} = \frac{ \sqrt{6}+ \sqrt{2} }{4}
To find \tan 75^\, , you can use the tangent addition formula with angles 45^\, + 30^\, . After substituting the values associated with those angles, you will derive the exact value as 3 − 1 3 + 3 .
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