How do you find the exact value of arctan(2-√3)
so we need an angle θ , so that
tan θ = 2 - √3
( we could be sneaky and use our calculator, sure enough it says 15°)
so we will make an "intelligent guess", then prove my guess is correct
we know tan 30° = 1/√3
and I also know that tan 30° = 2 tan15°/(1 - tan^2 15°)
let tan15° = x
so tan30° = 2 tan15/(1 - tan^2 15°) becomes
1/√3 = 2x/(1-x^2)
2√3 x = 1 - x^2
x^2 + 2√3 x - 1 = 0
x = ( -2√3 ± √(12+4) )/2
= -√3 ± 2 <----- well, who would have guessed!
but we know x = tan 15
so 2 - √3 = tan 15° , and
arctan(2-√3) = 15° or π/12 radians
OR
Multiply and divide 2 - √3 by 1/ 2
2 - √3 = [ ( 2 - √3 ) / 2 ] / ( 1 / 2 ) =
( 1 - √3 / 2 ) / ( 1 / 2 )
Since:
sin 30°= 1 / 2
cos 30° =√3 / 2
and half-angle formula for tangent is:
tan ( θ / 2 ) = ( 1 - cos θ ) / sin ( θ )
tan ( 30° / 2 ) =
( 1 - cos 30° ) / sin 30 °=
( 1 - √3 / 2) / ( 1 / 2 ) = tan 15°
arctan ( 2 - √3 ) =
arctan ( 1 - √3 / 2) / ( 1 / 2 ) =
15° = π / 12 rad
To find the exact value of arctan(2-√3), we can use the properties of right triangles and the unit circle. Here are the steps to find the answer:
Step 1: Start by drawing a right triangle in the coordinate plane. Let's call the angle at the origin "θ", and label the side adjacent to θ as "a" and the side opposite to θ as "b".
Step 2: Based on the given value, we have that tan(θ) = 2-√3. Since tan(θ) is the ratio of the opposite side (b) to the adjacent side (a), we can write the following equation: (2-√3) = b/a.
Step 3: We can square both sides of the equation to eliminate the square root: (2-√3)^2 = (b/a)^2, which simplifies to 7 - 4√3 = (b/a)^2.
Step 4: Since we want to find arctan(2-√3), we are looking for the angle θ whose tangent equals 2-√3. So, we need to find the ratio b/a.
Step 5: To solve for b/a, we can take the square root of both sides of the equation found in step 3: √(7 - 4√3) = b/a.
Step 6: Using a calculator, we can evaluate the square root of (7 - 4√3) to get an approximate decimal value. Let's call it x.
Step 7: Now we have b/a = x, which means b = x*a.
Step 8: Using the Pythagorean theorem, we have a^2 + b^2 = 1. Substituting b = x*a, we get a^2 + (x*a)^2 = 1.
Step 9: Solving the equation from step 8, we find the value of a.
Step 10: Once we know the values of a and b, we can find the angle θ by using the inverse tangent function. Therefore, the exact value of arctan(2-√3) is the angle θ (in radians) corresponding to a/b.
Please note that steps 6-9 involve some calculations that may require a calculator.