Problem 16 - Algebra 2

Question

Find the exact value of each expression. If the expression is undefined, write undefined. $$ \cot 30^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \(\cot(30^{\circ})\) is \(\sqrt{3}\).

Step 1: Understanding Cotangent

The cotangent of an angle can be obtained from the tangent of the same angle, as \(\cot(\theta) = \frac{1}{\tan(\theta)}\). In this example, the angle is 30 degrees.

Step 2: Using Tangent Values

The value of \(\tan(30^{\circ})\) is \(\frac{\sqrt{3}}{3}\). Now substitute this value to find cotangent.

Step 3: Solving for Cotangent

Substitute the value of \(\tan(30^{\circ})\) into the cotangent expression, we get \(\cot(30^{\circ}) = \frac{1}{\tan(30^{\circ})} = \frac{1}{\frac{\sqrt{3}}{3}} = \sqrt{3}\)

Key Concepts

Cotangent

The cotangent of an angle is a trigonometric function that relates to the tangent of the same angle. In simple terms, cotangent is the reciprocal of the tangent. To find the cotangent, you simply take one divided by the tangent of the angle. This can be expressed with the formula:\[\cot(\theta) = \frac{1}{\tan(\theta)}\]So, if we're dealing with an angle such as 30 degrees, we first find the tangent of this angle. Then, we take its reciprocal to get the cotangent value. Cotangent is an important function in trigonometry and is frequently used in solving right triangles. In our exercise, once we know \(\tan(30^\circ)\) is \(\frac{\sqrt{3}}{3}\), finding \(\cot(30^\circ)\) becomes straightforward by calculating:\[\cot(30^\circ) = \frac{1}{\frac{\sqrt{3}}{3}} = \sqrt{3}\]

Tangent

Tangent is one of the primary trigonometric functions and is essential for understanding cotangent. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. This is often remembered by the acronym "SOHCAHTOA," where the "TOA" part is for Tangent = Opposite/Adjacent.Knowing the tangent helps in finding other trigonometric values, just like in the case of cotangent. For standard angles, such as 30 degrees, the tangent has known values that can be found in trigonometric tables or remembered from common patterns. For example, \(\tan(30^\circ)\) equals \(\frac{\sqrt{3}}{3}\). Having a clear understanding of these standard values can greatly ease solving trigonometric expressions.Tangent and cotangent are closely linked, and understanding this relationship helps in simplifying complex trigonometric problems.

Exact Value of Trigonometric Expressions

Finding the exact value of trigonometric expressions is a key skill in mathematics. Trigonometric values can be expressed as exact numbers rather than decimal approximations, which is crucial for precise calculations and solving equations.

• Standard angles like 30°, 45°, and 60° often have known trigonometric values.
• For example, for 30°: \(\sin(30^\circ) = \frac{1}{2}\), \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), and \(\tan(30^\circ) = \frac{\sqrt{3}}{3}\).

These exact values are derived from either special right triangles—like the 30-60-90 triangle—or the unit circle. They allow us to solve equations without needing calculators. Instead of rounding, these expressions give us more accurate results.When solving the cotangent in the exercise, knowing the exact tangent value for 30° was essential to find the precise result of \(\sqrt{3}\). This approach ensures accuracy and reliability in mathematical problem-solving.