Problem 13 - Algebra 2 and Trigonometry

Question

For each function value, write the value or tell why it is undefined. Do not use a calculator. \(\tan 0\)

Step-by-Step Solution

Answer

\( \tan 0 = 0 \)

Step 1: Understanding tangent function

The tangent function, denoted as \( \tan \theta \), is the ratio of the opposite side to the adjacent side in a right-angled triangle. Alternatively, it can be described in terms of sine and cosine as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Step 2: Substitute angle value

In the given problem, we need to find \( \tan 0 \). Substitute \( \theta = 0 \) into the tangent function: \( \tan 0 = \frac{\sin 0}{\cos 0} \).

Step 3: Calculate sine and cosine for 0 degrees

Recall that \( \sin 0 = 0 \) and \( \cos 0 = 1 \). These are standard values for the sine and cosine functions at 0 degrees (or 0 radians).

Step 4: Perform the division

Substitute the values from Step 3 into the expression from Step 2: \( \tan 0 = \frac{0}{1} \). Calculate the division to get \( 0 \).

Key Concepts

Trigonometric Functions

Trigonometric functions are a crucial part of mathematics, especially in geometry. They help us understand the relationships within a right-angled triangle. There are six primary functions: sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)). Usually, these functions correlate angles to ratios that describe the triangle's sides.

In most trigonometric functions, angles can be expressed in degrees or radians. For example, a full circle is 360 degrees or \(2\pi\) radians. Understanding these functions is fundamental for solving many mathematical problems involving triangles and circles. Trigonometric functions expand beyond simple triangles to model periodic behaviors, such as waves or oscillations.

Sine and Cosine

The sine and cosine functions are essential tools in trigonometry. They define fundamental relationships between the angles and lengths of a right-angled triangle. The sine function \(\sin(\theta)\) represents the ratio of the opposite side to the hypotenuse. In contrast, the cosine function \(\cos(\theta)\) is the ratio of the adjacent side to the hypotenuse.

When working with angles of zero degrees, these functions are straightforward:

• \(\sin 0 = 0\)
• \(\cos 0 = 1\)

These values are a foundation for understanding other trigonometric relationships. Sine and cosine also serve as building blocks for the tangent function: \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\). Calculating the values for sine and cosine at various angles allows us to understand the behavior of the tangent function better.

Undefined Values in Trigonometry

In trigonometry, some values of trigonometric functions can be undefined. This mainly occurs with tangent, secant, cosecant, and cotangent functions due to division by zero issues.

A prime example is the tangent function \(\tan(\theta)\). Recall \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\). If the cosine of the angle (\(\cos(\theta)\)) is zero, the tangent function becomes undefined because division by zero is not possible.

• For example, \(\tan(90^\circ)\) is undefined because \(\cos(90^\circ) = 0\).

When solving problems in trigonometry, it's crucial to recognize when these functions are undefined. This ensures that calculations remain valid and help avoid mistakes in mathematical operations. Understanding why certain trigonometric values are undefined allows students to better navigate through more complex trigonometric identities and equations.