SOLUTION: Draw an equilateral triangle and bisect it. Prove that the exact value of the tangent ratios oif 30 degree, 45 degree, and 60 degree are 0.577, 1, and 1.73 respectively. Kindly

Question 206932This question is from textbook Longman Mathematics for IGCSE book1
: Draw an equilateral triangle and bisect it. Prove that the exact value of the tangent ratios oif 30 degree, 45 degree, and 60 degree are 0.577, 1, and 1.73 respectively.
Kindly solve this question. It is really urgent for me!
Thankyou very much indeed. This question is from textbook Longman Mathematics for IGCSE book1

Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
Draw an equilateral triangle and bisect it. Prove that the exact value of the tangent ratios of 30 degree, 45 degree, and 60 degree are 0.577, 1, and 1.73 respectively.
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Draw an equilateral triangle with sides of length x,x,x.
Draw a bisector of one of the angles.
This will also be a bisector of the opposite side and will form 2 rt. angles.
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You now have two right triangles with angles of 30, 60 and 90 degrees
and sides of x,(x/2), and ?
Solve for ? using Pythgoras:
x^2 = (x/2)^2 + ?^2
?^2 = (3/4)x^2
? = [(sqrt(3))/2]x
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So tan(60 deg) = ?/(x/2) = [(sqrt(3))/2]x/(x/2) = sqrt(3) = 1.73 when rounded.
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Also tan(30) = (x/2)/[(sqrt(3))/2]x = 1/sqrt(3) = 0.577 when rounded.
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I don't see any 45 degree angles in the equilateral figure.
But if you draw a right triangle with two equal legs
the tan(45) = x/x = 1.
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Cheers,
Stan H.
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