Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-38,\) find each function value to four decimal places. $$ \tan 375^{\circ} $$

Step-by-Step Solution

Answer

\( \tan 375^\circ = 0.2679 \).

Step 1: Conceptualize the Tangent Function

The tangent function, \( \tan \theta \), is periodic with a period of \(360^\circ \). This means that \( \tan(\theta) = \tan(\theta + 360^\circ) \). Knowing this, we can find equivalent angles for a given angle that differ by multiples of \(360^\circ \).

Step 2: Reduce the Angle

Given the angle \( 375^\circ \), use the periodicity property: Subtract \( 360^\circ \) from \(375^\circ\) to reduce the angle within the standard range \([0^\circ, 360^\circ)\). Thus, \[ 375^\circ - 360^\circ = 15^\circ. \] Therefore, \( \tan 375^\circ = \tan 15^\circ \).

Step 3: Calculate Tangent of the Reduced Angle

Now calculate \( \tan 15^\circ \). Utilize known trigonometric values or a calculator. Using a calculator, \( \tan 15^\circ \approx 0.2679 \).

Step 4: Answer Verification

Verify the calculated tangent value by rechecking with a calculator. Ensure the calculator is set to degree mode to avoid any discrepancies. Confirm \( \tan 15^\circ \approx 0.2679 \) is correct.

Key Concepts

Periodicity of Tangent Function

The tangent function, denoted as \(\tan(\theta)\), displays a characteristic known as periodicity. This means its values repeat after a certain interval, helping us simplify calculations. For the tangent function, this period is \(360^\circ\). While sine and cosine have periods of \(360^\circ\) and \(180^\circ\) respectively, tangent aligns with sine and cosine over a complete circle of \(360^\circ\). Understanding this periodicity simplifies complex angle problems by allowing reduction of angles that extend beyond the standard circle measurement.

• This properties means that \( \tan(\theta) = \tan(\theta + 360^\circ) \).
• By knowing this, angles larger than \(360^\circ\) can be adjusted into a simpler, equivalent angle for easier computation.

This periodic attribute allows students to relate practically any angle to positions within a circle, refining calculations and ensuring the solutions are both quicker and more straightforward.

Trigonometric Calculations

Trigonometric calculations are integral in solving problems involving angles. The key functions include sine, cosine, and tangent. Each function provides unique values that represent ratios of different sides of a right triangle. When tackling problems similar to finding \(\tan 375^\circ\), understanding these functions plays a crucial role. For performing these calculations, it's essential to have the angle in degrees or radians. Ensure your calculator is set accordingly to prevent mistakes. With right reasoning and steps, trigonometric calculations can become second nature.

• Explore known values: Some angles like \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), and \(90^\circ\) have known tangent values that make rapid calculations possible.
• Use calculators proficiently: They assist in confirming values when angles fall outside common memorized benchmarks.

Mastering these calculations is a strong foundation for more advanced studies in mathematics and physics.

Angle Reduction Technique

Angle reduction is a valuable technique used when dealing with larger angles like \(375^\circ\). It involves transforming these angles into more manageable equivalents fitting within a standard range \([0^\circ, 360^\circ)\). This is achieved predominantly by subtracting multiples of \(360^\circ\) until the angle falls within this standard range, ensuring simplified calculations and accurate results.

• The reduced angle retains the same trigonometric values since they share a common reference on the unit circle.
• For example, \(375^\circ - 360^\circ = 15^\circ\), so \(\tan 375^\circ = \tan 15^\circ\).

Utilizing this technique makes working through trigonometric problems more accessible and can be applied across various mathematical contexts, aiding in a smoother learning journey.