Finding the Value of tan(π/3) and Understanding the Tangent Function in Trigonometry

tan(π/3)

To find the value of tan(π/3), we first need to understand what the tangent function represents

To find the value of tan(π/3), we first need to understand what the tangent function represents. In trigonometry, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side in a right triangle.

Now, let’s consider the angle π/3 (pi/3 radians), which is equivalent to 60 degrees. We can construct a right triangle with an angle of 60 degrees. In this triangle, the opposite side to the angle π/3 (60 degrees) has length “y” and the adjacent side has length “x.”

Using the Pythagorean theorem, we can find the length of the hypotenuse “h” of our triangle. The Pythagorean theorem states that the sum of the squares of the lengths of the two legs of a right triangle is equal to the square of the length of the hypotenuse.

In our triangle, we have:
x^2 + y^2 = h^2

Since we want to find the tangent of π/3, we are interested in the ratio y/x. Notice that y/x is equivalent to the slope of the line defined by the right triangle sides. Tangent is defined as rise over run.

To find the values of x, y, and h in our triangle, we can use a unit circle. For the angle of π/3 (60 degrees), if we draw a radius from the origin of the unit circle, it will intersect the unit circle at a point with coordinates (1/2, √3/2).

Therefore, x = 1/2 and y = √3/2. We can now substitute these values into our equation to find h:

(1/2)^2 + (√3/2)^2 = h^2
1/4 + 3/4 = h^2
4/4 = h^2
1 = h^2

Taking the square root of both sides, we find that h = 1.

Now that we have the values of x, y, and h, we can calculate the tangent of π/3:

tan(π/3) = y/x = (√3/2) / (1/2)

To divide fractions, we multiply the numerator by the reciprocal of the denominator:

tan(π/3) = (√3/2) * (2/1) = √3

Therefore, the value of tan(π/3) is √3 or approximately 1.732.

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