Question

Use the fact that the trigonometric functions are periodic to find the exact value of each expression. Do not use a calculator. $$ \tan 405^{\circ} $$

Short answer

The value of \tan 405° is 1.

Step-by-step solution

Identify the Periodicity of the Tangent Function

The tangent function is periodic with a period of 180°. This means that \(\tan(\theta + 180°) = \tan(\theta)\) for any angle \( \theta \)

Simplify the Angle Using the Period

Since 405° is greater than 180°, reduce it within one period of the tangent function. \[ 405° - 2 \times 180° = 405° - 360° = 45° \]

Evaluate the Simplified Angle

Now, find \( \tan 45° \), which is equal to 1. Thus, \( \tan 405° = \tan 45° = 1 \)

Key concepts

Periodicity

Periodicity is a fundamental concept in trigonometry. Trigonometric functions repeat their values in regular intervals. For the tangent function, this periodicity is every 180°. This means that any angle can be reduced by multiples of 180° to find an equivalent angle within a standard range.

• For example, \(\tan(\theta + 180°) = \tan(\theta)\) applies to any angle \( \theta \).
• This makes calculations simpler because it lets us handle large angles by reducing them to smaller ones.

This property is particularly useful when evaluating angles that extend beyond the typical range of 0° to 360°. By recognizing periodicity, we can transform unmanageable angles into familiar and more easily solvable ones.

Tangent Function

The tangent function is one of the primary trigonometric functions, alongside sine and cosine. It is defined as the ratio of the sine to the cosine of an angle:
\(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).

• The function's behavior is periodic with a period of 180°.
• It is undefined when the cosine of the angle is zero since division by zero is undefined.

For angles like 90° and 270°, where cosine is zero, the tangent function is undefined. This function changes between positive and negative values as the angle progresses through its period, making it essential to understand its graph and periodicity.

Angle Reduction

Angle reduction involves simplifying angles to fall within a certain range by using the periodicity of the trigonometric functions. For the tangent function, this means reducing any angle to within 0° to 180°.

• In our example, \(\tan 405°\) was handled by noting that the tangent function has a period of 180°.
• We reduced \(405°\) by subtracting \(2 \times 180°\) (or 360°) which gives us 45°.

Reducing angles significantly simplifies the problem, enabling us to work with known values and avoid confusion. This process is key in solving trigonometric problems and finding exact values for various angles.

Exact Value Calculation

Exact value calculation means determining the exact trigonometric value without relying on a calculator. In the case of \(\tan 405°\), after reducing the angle, we calculate the tangent precisely.

• For \(\tan 45°\), by knowing standard trigonometric values, we determine that the exact value is 1, since \(\tan 45° = 1\).
• These exact values are typically derived from standard trigonometric identities and properties known for specific angles such as 30°, 45°, and 60°.

Understanding exact value calculation is critical as it helps in developing problem-solving skills without resorting to technological aids like calculators. This method encourages a deeper grasp of trigonometric principles.