Prove :Tan20+4sin20=square root 3
Let ABC be a right-angled triangle with ∠ABC = 60° and ∠CAB = 30°.
Let ABD be a right-angled triangle on the same side of AB
with ∠ABD = 40° and ∠DAB = 50°
Suppose that AC and BD intersect at E, and
that the length of BC is 1, so that the respective lengths of CA and AB are √3 and 2. Then
|AD| = |AB|sin 40° = 4 sin 20°cos 20° , (recall sin 2x = 2sinx cosx)
and
|AE| = |AD|sec 20° = |AB| cos 50°
sec 20° = 2 sin 40°
sec 20° = 4 sin 20°
However, |CE| = |BC|tan 20° = tan 20°
Therefore
tan 20° + 4 sin 20° = |CE| + |AE| = |AC| = √3
source:
http://www.math.toronto.edu/barbeau/olymon2010.pdf
page 9
To prove the equation, we will need to simplify both sides and show that they are equal.
Starting with the left side of the equation:
Tan(20) + 4sin(20)
First, we need to convert the tangent function to sine and cosine functions using the identity:
Tan(x) = sin(x) / cos(x)
Substituting the identity in the equation:
sin(20)/cos(20) + 4sin(20)
To combine these terms, we find a common denominator for the two terms on the left side:
(sin(20) + 4cos(20)sin(20))/cos(20)
Now, we can simplify the numerator by factoring out sin(20):
sin(20)(1 + 4cos(20))/cos(20)
Next, we can use the Pythagorean identity:
sin^2(x) + cos^2(x) = 1
Rearranging the equation, we have:
cos^2(x) = 1 - sin^2(x)
Substituting into our expression:
sin(20)(1 + 4(1 - sin^2(20)))/cos(20)
Now, we can simplify further:
sin(20)(1 + 4 - 4sin^2(20))/cos(20)
sin(20)(5 - 4sin^2(20))/cos(20)
Using the double-angle formula:
sin(2x) = 2sin(x)cos(x)
We can rewrite the expression as:
sin(20)(5 - 4sin(20)sin(20))/cos(20)
Simplifying:
sin(20)(5 - 4sin^2(20))/cos(20)
Using the identity:
cos(2x) = cos^2(x) - sin^2(x)
We can rewrite the expression as:
sin(20)(5 - 4(1 - cos^2(20)))/cos(20)
Expanding:
sin(20)(5 - 4 + 4cos^2(20))/cos(20)
Further simplification:
sin(20)(1 + 4cos^2(20))/cos(20)
Now, using the Pythagorean identity again:
sin^2(x) + cos^2(x) = 1
cos^2(x) = 1 - sin^2(x)
We can rewrite the expression as:
sin(20)(1 + 4(1 - sin^2(20)))/cos(20)
Expanding and simplifying:
sin(20)(5 - 4sin^2(20))/cos(20)
sin(20)(5) - sin(20)(4sin^2(20))/cos(20)
simplifying further:
5sin(20) - 4sin^3(20)/cos(20)
Using the identity:
sin(x)/cos(x) = tan(x)
We can rewrite the expression as:
5sin(20) - 4sin^3(20)tan(20)
Finally, we can substitute this result back into the original equation:
5sin(20) - 4sin^3(20)tan(20) = √3
Solving for the value of θ in the equation will give us the angle θ for which the equation holds.