Question

Find the exact value of each expression. Do not use a calculator. $$ 4+\tan ^{2} \frac{\pi}{3} $$

Short answer

7

Step-by-step solution

Recall the Tangent Value

First, recall that \(\tan \frac{\frac{\text{π}}{3}})\) = \(\tan 60^\text{°}})\) is equal to \(\text{√3}})\).

Square the Tangent Value

Now, find \((\tan \frac{\frac{\text{π}}{3}})^2})\) which is equal to \((\text{√3})^2})\). Thus, \[ (\text{√3})^2 = 3 \].

Add 4 to the Squared Tangent Value

Finally, add 4 to the squared tangent value: \(\text{4 + 3}})\), which equals \[ \text{7} \].

Key concepts

The Tangent Function

The tangent function, abbreviated as \(\tan\), is one of the basic trigonometric functions. It relates an angle in a right triangle to the ratio of the length of the opposite side to the adjacent side. In a unit circle, \(\tan\theta \) is the length of the vertical segment of a line through the origin that intersects the circle. This function is periodic with a period of \(\text{π}\ \text{radians}\). To understand it better, remember the following key properties: the function is undefined at \(\frac{\text{π}}{2}\ + \text{nπ}\), where \(\text{n}\) is an integer, and \(\tan \text{0°} = 0\).

Understanding π Radians

In trigonometry, angles can be measured in degrees or radians. Using radians simplifies many mathematical expressions, especially in calculus. One full circle is equivalent to \(\text{2π}\) radians, which is about \(\text{6.2832}\) radians. This means that \(\text{π}\) radians equal \(\text{180°}\). So when we talk about \(\frac{\text{π}}{3}\) radians, we are referring to \(\text{60°}\). Knowing the conversion between degrees and radians is crucial as it allows you to use trigonometric functions more effectively across different mathematical problems.

Square of Tangent

When you square the tangent of an angle, you essentially take the ratio found by the tangent function and multiply it by itself. In this exercise, knowing the value of \(\tan \frac{\text{π}}{3}\), which is \(\text{√3}\), and squaring it, we get: \[\tan ^{2} \frac{\text{π}}{3} = (\text{√3})^2 = 3\]. Squaring trigonometric functions often appears in trigonometric identities and can simplify solving equations. For instance, in this specific problem, squaring \(\tan \frac{\text{π}}{3}\) and adding 4 gives us \(\text{4 + 3 = 7}\), which is our final answer.