Understanding the Value of Tan 70° and Its Applications

The tangent function, denoted as tan, is a trigonometric function that provides the ratio of the sine to the cosine of an angle. In the context of a right triangle, it represents the ratio of the opposite side to the adjacent side. One specific value of interest is that of tan 70°, which has practical applications across various fields of study and practical scenarios. The value of (tan 70^circ) is approximately 2.7475. This value can be derived from various trigonometric relationships and identities, as we will explore in this article.

Deriving the Value of Tan 70°

To understand the value of (tan 70^circ) more deeply, letrsquo;s revisit the concept using basic trigonometric principles. We start with the well-known values of sine and cosine for certain special angles. For instance, we know that (sin 60^circ frac{sqrt{3}}{2}) and (cos 60^circ frac{1}{2}). From these, we can calculate (tan 60^circ) using the formula:

[tan theta  frac{sin theta}{cos theta}]

Substituting for (60^circ), we get:

[tan 60^circ  frac{frac{sqrt{3}}{2}}{frac{1}{2}}  sqrt{3}]

But to find (tan 70^circ), we don't have a direct formula. Instead, we use a known approximation, which gives us a rough value of (1.973), but for precise calculations, we often rely on calculators or trig tables.

Using Formulas and Approximations

Another approach to find the value of (tan 70^circ) is to use specific trigonometric identities. One such method involves using the triple angle formula for tangent:

[tan 3theta  frac{3tan theta - tan^3 theta}{1 - 3tan^2 theta}]

Setting (theta 35^circ), we want to find (tan 70^circ). Therefore, we can write:

[tan 70^circ  frac{3tan 35^circ - tan^3 35^circ}{1 - 3tan^2 35^circ}]

However, this requires us to know the value of (tan 35^circ), which can be tricky to compute by hand. An easier method for many is to use a known identity for double angles:

[tan (45^circ   30^circ)  frac{tan 45^circ   tan 30^circ}{1 - tan 45^circ tan 30^circ}]

Given that (tan 45^circ 1) and (tan 30^circ frac{1}{sqrt{3}}), we substitute these values in:

[tan (45^circ   30^circ)  frac{1   frac{1}{sqrt{3}}}{1 - 1 cdot frac{1}{sqrt{3}}} frac{1   frac{sqrt{3}}{3}}{1 - frac{sqrt{3}}{3}}]

Instead, we simplify:

[ frac{3   sqrt{3}}{3 - sqrt{3}}]

Rationalizing the denominator, we get:

[ frac{(3   sqrt{3})^2}{(3 - sqrt{3})(3   sqrt{3})}  frac{9   6sqrt{3}   3}{9 - 3}  frac{12   6sqrt{3}}{6}  2   sqrt{3}]

Now, we can further use similar triangulations with 35 degrees to solve for (tan 70^circ).

An Important Note on Calculations

Directly calculating (tan 70^circ) using these identities without a calculator can be quite involved. However, these methods illustrate the depth of trigonometric relationships and identities. For precise values, especially in applications, calculators or trigonometric tables are often used. The exact value of (tan 70^circ) is approximately 2.7475, which is crucial in fields such as engineering, physics, and architecture where precision matters.

Applications of Tan 70°

The value of (tan 70^circ) has practical applications in various fields. For instance, in engineering, it helps in calculating the angles and forces in structural designs. In physics, it aids in determining the trajectory and force of projectiles. In architecture, it is used to compute the angles and lengths in building designs.

Conclusion

The value of (tan 70^circ) is a fundamental trigonometric concept that finds its importance in comprehensive studies of mathematics and its applications. While precise values are often obtained using calculators or tables, understanding the foundational methods like trigonometric identities provides a deeper insight into the nature of trigonometric functions.