what is the exact value of tan375degrees
tan(375°)
=tan(2π+15°)
=tan(15°)
Since tan(30°)=(√3)/2,
we use
tan(θ/2)=(1-cosθ)/sinθ, or
tan(θ/2)=sinθ/(1+cosθ)
Thus
tan(15°)
=(1-cos(30°))/sin(30°)
=(1-√3/2)/(1/2)
=2-√3
haha that would be funny, but really there are two of us here (:
http://www.google.com/search?hl=en&rlz=1G1GGLQ_ENUS374&&sa=X&ei=RVcMTO7-HoL88AblxfyKBw&ved=0CB4QBSgA&q=tan+375+degrees&spell=1
no, i need an exact answer
tan(375 degrees) = 0.267949192
Did you click on the above link?
tan(375 degrees) = 0.267949192
yes, but that is not an exact answer. i need a fraction of some sort, not a rounded decimal
if that confused you, haley and i are working together.
To find the exact value of tan(375 degrees), we can use the periodicity of the tangent function. The tangent function has a period of 180 degrees, which means it repeats every 180 degrees.
First, we need to determine the reference angle. To do this, we subtract a multiple of 180 degrees from the given angle to bring it within the range of 0 to 180 degrees.
375 degrees - 360 degrees = 15 degrees
Since 15 degrees is within the first quadrant (0 to 90 degrees), the tangent function will be positive.
Next, we find the exact value of tangent for the reference angle of 15 degrees.
To do this, we can use the exact values of trigonometric functions for commonly found angles (like 0, 30, 45, 60, 90 degrees) and the different trigonometric identities.
However, for an angle of 15 degrees, there are no exact values that we can use directly. In this case, we can use the angle addition formula for tangent, which states that tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a) * tan(b)).
We can rewrite 15 degrees as the sum of two angles, 45 degrees and -30 degrees:
15 degrees = 45 degrees - 30 degrees
tan(15 degrees) = tan(45 degrees - 30 degrees)
Now, we can use the angle addition formula to evaluate the tangent function:
tan(45 degrees - 30 degrees) = (tan(45 degrees) + tan(-30 degrees)) / (1 - tan(45 degrees) * tan(-30 degrees))
tan(45 degrees) = 1 (exact value)
tan(-30 degrees) = -tan(30 degrees) = -√3/3 (exact value)
Substituting these values into the formula:
tan(15 degrees) = (1 + (-√3/3)) / (1 - 1 * (-√3/3))
Simplifying:
tan(15 degrees) = (1 - √3/3) / (1 + √3/3)
To get a precise value, we can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator:
tan(15 degrees) = [(1 - √3/3) * (3 + √3)] / [(1 + √3/3) * (3 + √3)]
tan(15 degrees) = (3 - √3 + 3√3 - 3)/ (3 + √3 + 3√3 + √3)
Simplifying further:
tan(15 degrees) = (6√3 - √3) / (3 + 4√3)
tan(15 degrees) = 5√3 / (3 + 4√3)
So, the exact value of tan(375 degrees) is 5√3 / (3 + 4√3).