tan 15°
tan (45°-30°)
(tan 45° - tan 30°)/1+ tan 45°tan30°
(1-√3/3)/(1+1√3/3)
then i donnt what to do/ chancel out. Can someone finish it if i didn't get it wrong.
thanks in advance
I would rationalize the denominator
(1-1/3 sqrt3)^2/(1-1/3)=3(1-1/3 sqrt3)^2/2
To find the value of the given expressions involving trigonometric functions, we'll need to use some trigonometric identities and simplify the expressions step by step.
1. tan 15°:
To find the value of tan 15°, we can use the half-angle formula for tangent:
tan(θ/2) = ± √((1 - cosθ) / (1 + cosθ))
In this case, θ = 30°, so we have:
tan 15° = ± √((1 - cos 30°) / (1 + cos 30°))
Now, let's find the value of cos 30° using a common trigonometric identity:
cos 30° = √3 / 2
Substituting the value of cos 30° into the equation, we get:
tan 15° = ± √((1 - √3 / 2) / (1 + √3 / 2))
2. tan (45° - 30°):
We can simplify this expression by using the sum-to-product identity for tangent:
tan (A - B) = (tan A - tan B) / (1 + tan A * tan B)
In this case, A = 45° and B = 30°, so we have:
tan (45° - 30°) = (tan 45° - tan 30°) / (1 + tan 45° * tan 30°)
Now, substitute the values of tan 45° and tan 30°:
tan (45° - 30°) = (1 - √3/3) / (1 + 1√3/3)
3. (tan 45° - tan 30°) / (1 + tan 45°tan30°):
We already have the values of tan 45° and tan 30°:
(1 - √3/3) / (1 + 1√3/3)
To simplify this expression further, we can multiply both the numerator and denominator by the conjugate of the denominator to eliminate the radical in the denominator. The conjugate of 1 + 1√3/3 is 1 - 1√3/3.
Multiplying the numerator and denominator by the conjugate, we get:
[(1 - √3/3) * (1 - 1√3/3)] / [(1 + 1√3/3) * (1 - 1√3/3)]
Simplifying the expression:
[(1 - √3/3)(1 - 1√3/3)] / (1 - √3/3)²
Expanding the numerator and simplifying:
[(1 - √3/3 - √3/3 + 3/9)] / (1 - (2√3/3) + (3/9))
Continuing to simplify:
[(1 - 2√3/3 + 3/9)] / (1 - 2√3/3 + 1/3)
Now, combine like terms in the numerator:
[(4 - 2√3)/9] / (1 - 2√3/3 + 1/3)
Finally, simplify the expression by combining like terms in the denominator:
(4 - 2√3) / (9 - 6√3 + 3)