Understanding arctan 4 3 in mathematics

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arctan 4 3
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The inverse tangent function arctan 4 3 represents a fundamental angle in trigonometry whose geometric and computational significance extends across disciplines. As the angle whose tangent equals 4/3, it emerges naturally in right triangles, slope calculations, and advanced integration techniques. This exploration dissects its exact value, numerical approximations, and practical applications while contrasting it with complementary angles like arctan 3 4.

From geometric constructions using compass and straightedge to algorithmic implementations in programming languages, arctan 4 3 serves as a bridge between theoretical mathematics and real-world problem-solving. Its precise derivation through Taylor series expansions and inverse trigonometric identities underscores its role in both analytical and computational frameworks. Additionally, visual representations—such as unit circle diagrams and dynamic right-triangle animations—reveal deeper insights into its trigonometric relationships and hyperbolic reflections.

arctan 4 3

Mathematical Definition and Properties of arctan(4/3)

The inverse tangent function, denoted as arctan(x), computes the angle whose tangent is x within the range \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians. For the specific case of arctan(4/3), the value represents the angle in a right triangle where the opposite side is 4 units and the adjacent side is 3 units. This geometric interpretation is foundational in trigonometry, linking algebraic expressions to visual representations of angles.

The exact and approximate values of arctan(4/3) are derived from its definition and can be expressed in both radians and degrees. Additionally, its computation via series expansion and relationships with other inverse trigonometric functions provide deeper insights into its mathematical behavior.

Geometric Interpretation in a Right Triangle

The value arctan(4/3) corresponds to an angle θ in a right triangle where:
  • The opposite side (to θ) is 4 units.
  • The adjacent side (to θ) is 3 units.
  • The hypotenuse can be computed using the Pythagorean theorem as \( \sqrt{3^2 + 4^2} = 5 \) units.

    • This configuration is a 3-4-5 Pythagorean triple, a well-known right triangle with integer side lengths. The angle θ = arctan(4/3) is acute, lying between 0 and π/2 radians (0° and 90°).

    The trigonometric ratios for this angle are:

  • sin(θ) = 4/5
  • cos(θ) = 3/5
  • tan(θ) = 4/3 (by definition).

    • These ratios are consistent with the unit circle and standard trigonometric identities.

    Exact and Approximate Values in Radians and Degrees

    The exact value of arctan(4/3) cannot be expressed in terms of elementary functions (e.g., π, square roots) using standard algebraic operations. However, it can be represented numerically or in terms of inverse trigonometric identities.

    Approximate values:

  • In radians: arctan(4/3) ≈ 0.9272952180016122 (≈ 0.9273 rad).
  • In degrees: arctan(4/3) ≈ 53.13010235415598° (≈ 53.13°).

    • These approximations are derived using computational tools (e.g., calculators or programming libraries) and are accurate to 15 decimal places. For practical purposes, rounding to 53.13° or 0.9273 rad is sufficient in most applications.

    Computation via Taylor Series Expansion

    The Taylor series expansion for arctan(x) centered at x = 0 is given by:
    \[
    \text{arctan}(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} - \cdots
    \]
    For x = 4/3, the series converges slowly due to the value being outside the interval \([-1, 1]\), where convergence is optimal. However, the first five non-zero terms provide a reasonable approximation:

    \[
    \text{arctan}\left(\frac{4}{3}\right) \approx \frac{4}{3} - \frac{(4/3)^3}{3} + \frac{(4/3)^5}{5} - \frac{(4/3)^7}{7} + \frac{(4/3)^9}{9}
    \]

    Step-by-step computation:
    1. First term: \( \frac{4}{3} \approx 1.3333333333333333 \)
    2. Second term: \( -\frac{(4/3)^3}{3} = -\frac{64/27}{3} = -\frac{64}{81} \approx -0.7901234567901234 \)
    3. Third term: \( \frac{(4/3)^5}{5} = \frac{1024/243}{5} \approx 0.8433909797531321 \)
    4. Fourth term: \( -\frac{(4/3)^7}{7} = -\frac{16384/2187}{7} \approx -0.1076607628112644 \)
    5. Fifth term: \( \frac{(4/3)^9}{9} = \frac{262144/19683}{9} \approx 0.0149011611904762 \)

    Summing the terms:
    \[
    1.3333333333333333 - 0.7901234567901234 + 0.8433909797531321 - 0.1076607628112644 + 0.0149011611904762 \approx 0.993940294665354
    \]
    This approximation (≈ 0.9939 rad) differs from the actual value (≈ 0.9273 rad) due to the slow convergence. For better accuracy, additional terms or alternative methods (e.g., Machin-like formulas) are required.

    Comparison Table of Inverse Tangent Values

    The following table contrasts arctan(4/3) with other common inverse tangent values, highlighting their relationships and approximate magnitudes:
    Function Exact Value (if applicable) Approximate Value (radians) Approximate Value (degrees) Trigonometric Context
    arctan(4/3) No closed-form expression 0.9273 53.13° Angle in a 3-4-5 right triangle
    arctan(3/4) No closed-form expression 0.6435 36.87° Complementary angle to arctan(4/3) in a 3-4-5 triangle
    arctan(1) π/4 ≈ 0.7854 0.7854 45° Standard angle in unit square
    arctan(√3) π/3 ≈ 1.0472 1.0472 60° Angle in an equilateral triangle
    Key Observations:
  • arctan(4/3) and arctan(3/4) are complementary angles in a 3-4-5 triangle, summing to π/2 (90°).
  • arctan(1) corresponds to 45°, a fundamental angle in trigonometry.
  • arctan(√3) equals 60°, another standard angle derived from equilateral triangles.

    • Exact Value Derivation Using Inverse Trigonometric Identities

    While arctan(4/3) lacks a simple closed-form expression, its exact value can be represented using inverse trigonometric identities, particularly those involving sums of arctangent functions. The following identities are relevant:

    1. Sum of Arctangents:
    \[
    \text{arctan}(x) + \text{arctan}\left(\frac{1}{x}\right) = \frac{\pi}{2} \quad \text{for } x > 0
    \]
    Applying this

    Applications of arctan(4/3) in Trigonometry and Geometry

    The inverse tangent function, arctan(4/3), frequently emerges in practical trigonometric and geometric problems where angles are defined by ratios of perpendicular to adjacent sides in right triangles. Its applications span from engineering design to navigation, where precise angle measurements are critical. This section explores its role in slope calculations, geometric constructions, parametric equations, and integration techniques, emphasizing its utility in both theoretical and applied contexts.

    Slope Angles and Real-World Applications

    The value arctan(4/3) represents the angle θ whose tangent is 4/3, a ratio commonly encountered in civil engineering, architecture, and road design. For instance, a road with a gradient of 4:3 (vertical rise to horizontal run) would have an inclination angle of arctan(4/3) ≈ 53.13°. This angle is significant in determining safe driving conditions, drainage slopes, and structural stability.

    In structural engineering, beams or trusses often form right triangles where the ratio of vertical to horizontal forces or dimensions may yield arctan(4/3). For example, a diagonal brace in a rectangular frame with a 4-unit vertical support and 3-unit horizontal span would create an angle of arctan(4/3) with the horizontal, influencing load distribution calculations. Similarly, in surveying, this angle may arise when measuring elevation changes over a known horizontal distance.

    Geometric Construction of arctan(4/3) Using Compass and Straightedge

    Constructing an angle of arctan(4/3) involves creating a right triangle with adjacent and opposite sides in the ratio 3:4. The following step-by-step procedure ensures geometric precision:

    1. Draw a Horizontal Baseline
    Use a straightedge to draw a horizontal line segment AB of arbitrary length (e.g., 3 units). This segment represents the adjacent side of the right triangle.

    2. Construct a Perpendicular at Point A
    At point A, use a compass to mark a point C such that AC is perpendicular to AB. Measure AC to be 4 units (the opposite side). Connect points B and C to form the hypotenuse.

    3. Verify the Ratio
    Confirm that the ratio of AC to AB is 4:3 by measuring both segments. The angle at B, ∠ABC, is arctan(4/3).

    4. Geometric Proof
    By the definition of tangent, tan(∠ABC) = AC/AB = 4/3. Thus, ∠ABC = arctan(4/3). The construction relies on the Pythagorean theorem to ensure the triangle is right-angled at A.

    Key Insight:
    The hypotenuse BC can be calculated as √(3² + 4²) = 5 units, confirming a 3-4-5 Pythagorean triple, a fundamental ratio in Euclidean geometry.

    Role in Parametric Equations and Polar Coordinates

    In parametric equations, arctan(4/3) often appears as an angle parameterizing curves or defining direction vectors. For example, a point moving along a line with slope 4/3 in Cartesian coordinates can be expressed parametrically as:
    x = 3t, y = 4t, where t ∈ ℝ.
    The angle θ of the line with the positive x-axis is arctan(4/3), simplifying calculations involving direction cosines or unit vectors.

    In polar coordinates, arctan(4/3) directly specifies the angular component (θ) when the radial distance (r) and Cartesian coordinates (x, y) satisfy:
    θ = arctan(y/x) = arctan(4/3).
    This relationship is critical in converting between Cartesian and polar forms, particularly in physics and engineering applications involving rotational symmetry.

    The inverse tangent function arctan(4/3) serves as a natural bridge between Cartesian coordinates (x, y) and polar angles θ, where θ = arctan(y/x) when x > 0. Its use in parametric equations ensures consistency in representing linear motion, rotational dynamics, and geometric transformations.

    Complementary Angles: arctan(4/3) vs. arctan(3/4)

    The angles arctan(4/3) and arctan(3/4) are complementary, meaning their sum is π/2 radians (90°). This relationship arises from the co-function identity:
    arctan(x) + arctan(1/x) = π/2, for x > 0.
    Substituting x = 4/3 yields:
    arctan(4/3) + arctan(3/4) = π/2.

    Practical Implications:

  • In trigonometric calculations, if an angle θ = arctan(4/3) is known, its complement (π/2 − θ) can be expressed as arctan(3/4).
  • This property is exploited in solving right triangles where one angle’s tangent is the reciprocal of another’s, such as in complementary slope calculations (e.g., road grades and their perpendicular counterparts).

    • Example:
    A road with a 4:3 gradient (θ = arctan(4/3)) has a complementary angle (π/2 − θ) representing the angle of its perpendicular bisector, which would be arctan(3/4).

    Trigonometric Substitution for Rational Functions

    The substitution θ = arctan(4/3) is instrumental in integrating rational functions involving √(x² + a²) or √(a² − x²). For instance, consider the integral:
    ∫ dx / √(x² + 9).

    Let x = 3tanθ, then dx = 3sec²θ dθ, and √(x² + 9) = 3secθ. Substituting yields:
    ∫ (3sec²θ dθ) / (3secθ) = ∫ secθ dθ = ln|secθ + tanθ| + C.

    To express the result in terms of x, use θ = arctan(x/3). If x/3 = 4/3 (i.e., x = 4), then θ = arctan(4/3), and the integral evaluates to:
    ln|sec(arctan(4/3)) + (4/3)| + C.

    Generalization:
    For integrals of the form ∫ dx / √(x² + a²), the substitution x = a tanθ transforms the integrand into secθ, leveraging the identity:
    1 + tan²θ = sec²θ.
    The angle θ = arctan(x/a) often simplifies to arctan(4/3) when x = 4a/3, demonstrating the function’s utility in analytical solutions.

    Trigonometric substitution via arctan(4/3) reduces complex rational integrals to standard forms, exploiting the Pythagorean identity and the properties of inverse trigonometric functions to yield closed-form solutions.

    arctan 4 3 - Ilustrasi 2

    Computational and Numerical Methods for Evaluating arctan(4/3)

    The evaluation of the inverse tangent function, particularly for non-standard arguments like \( \frac{4}{3} \), often requires computational or numerical methods when exact symbolic representations are impractical. These approaches leverage iterative algorithms, built-in mathematical libraries, or custom approximations to achieve precise results. Below, implementations in programming languages, numerical approximation techniques, and considerations for floating-point precision are explored.

    Implementation in Programming Languages

    Most high-level programming languages provide built-in functions to compute the arctangent of a value. For \( \arctan\left(\frac{4}{3}\right) \), these functions exploit optimized algorithms (e.g., CORDIC, Taylor series) to balance accuracy and performance.

    Python Example:

    import math
    result = math.atan(4/3) # Returns arctan(4/3) in radians
    print(f"arctan(4/3) ≈ {result:.10f} radians") # Output: ~0.9272952180 radians

    MATLAB Example:

    result = atan(4/3); % Returns arctan(4/3) in radians
    disp(['arctan(4/3) ≈ ', num2str(result, '%.10f'), ' radians']);

    Custom Approximation (Taylor Series):
    The Taylor series expansion for \( \arctan(x) \) around \( x = 0 \) converges for \( |x| \leq 1 \). For \( x = \frac{4}{3} \), a modified series or range reduction (e.g., \( \arctan(x) = \frac{\pi}{2} - \arctan\left(\frac{1}{x}\right) \) for \( x > 1 \)) is necessary.

    def custom_arctan(x, terms=10):
    x_reduced = x if abs(x) <= 1 else 1/x
    sign = -1 if x < 0 else 1
    result = 0.0
    for n in range(terms):
    term = ((-1)n) (x_reduced (2n + 1)) / (2n + 1)
    result += term
    return sign (result if abs(x) <= 1 else (math.pi/2 - result))

    print(f"Custom approximation: {custom_arctan(4/3):.10f}")

    Note: Convergence slows for \( |x| > 1 \); higher terms or adaptive methods improve accuracy.

    Numerical Approximation Methods

    When exact or library-based solutions are unavailable, iterative numerical methods approximate \( \arctan\left(\frac{4}{3}\right) \) by solving \( \tan(y) = \frac{4}{3} \). Below is a table summarizing key methods, including pseudocode and convergence criteria.
    Method Pseudocode Convergence Criteria Remarks
    Newton-Raphson
    1. Define \( f(y) = \tan(y) - \frac{4}{3} \), \( f'(y) = \sec^2(y) \).
    2. Initialize \( y_0 \) (e.g., \( \frac{\pi}{4} \)).
    3. Iterate: \( y_{n+1} = y_n - \frac{f(y_n)}{f'(y_n)} \).
    		 \( |y_{n+1} - y_n| < \epsilon \) (e.g., \( 10^{-6} \)). Fast convergence (quadratic) but requires derivative.
    Bisection
    1. Choose interval \( [a, b] \) where \( \tan(a) < \frac{4}{3} < \tan(b) \).
    2. Compute midpoint \( c = \frac{a+b}{2} \).
    3. Update interval: if \( \tan(c) < \frac{4}{3} \), set \( a = c \); else \( b = c \).
    		 \( b - a < \epsilon \). Guaranteed convergence but linear rate.
    Secant
    1. Initialize \( y_0, y_1 \) (e.g., \( 0, \frac{\pi}{2} \)).
    2. Iterate: \( y_{n+1} = y_n - f(y_n) \cdot \frac{y_n - y_{n-1}}{f(y_n) - f(y_{n-1})} \).
    		 \( |y_{n+1} - y_n| < \epsilon \). Superlinear convergence; avoids derivative computation.

    Binary Search Algorithm for arctan(4/3) with Precision \( 10^{-6} \)

    Binary search isolates \( y \) such that \( \tan(y) = \frac{4}{3} \) within a tolerance of \( 10^{-6} \). The algorithm exploits the monotonicity of \( \tan(y) \) in \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).

    Steps:
    1. Initial Bounds: Select \( a = 0 \), \( b = \frac{\pi}{2} \) (since \( \tan(0) = 0 < \frac{4}{3} < \infty = \tan\left(\frac{\pi}{2}^-\right) \)).
    2. Iteration:

  • Compute midpoint \( c = \frac{a + b}{2} \).
  • Evaluate \( \tan(c) \):
  • If \( \tan(c) < \frac{4}{3} \), set \( a = c \).
  • Else, set \( b = c \).
    • 3. Termination: Stop when \( b - a < 10^{-6} \). The solution \( y \approx \frac{a + b}{2} \).

    Pseudocode:

    import math

    def binary_search_arctan(target, tol=1e-6):
    a, b = 0.0, math.pi / 2
    while b - a > tol:
    c = (a + b) / 2
    if math.tan(c) < target:
    a = c
    else:
    b = c
    return (a + b) / 2

    result = binary_search_arctan(4/3)
    print(f"Binary search result: {result:.10f} radians")

    Output: Converges to \( \approx 0.927295 \) radians in ~20 iterations.

    Floating-Point Errors in arctan(4/3) Computation

    Floating-point arithmetic introduces rounding errors due to finite precision representations (e.g., IEEE 754 double-precision). For \( \arctan\left(\frac{4}{3}\right) \), these errors manifest in:
  • Truncation: Taylor series approximations truncate infinite sums, accumulating residual errors.
  • Machine Epsilon: Operations near \( \frac{\pi}{2} \) (where \( \tan(y) \to \infty \)) amplify errors due to limited exponent range.
  • Cancelation: Subtraction of nearly equal values (e.g., in range reduction) exacerbates precision loss.
    • The relative error in \( \arctan(x) \) for \( x \approx 1 \) is bounded by \( O(\epsilon) \), where \( \epsilon \) is the machine epsilon (~2.22e-16 for double-precision). For \( x = \frac{4}{3} \), errors propagate through intermediate steps (e.g., \( \sec^2(y) \) in Newton-Raphson), requiring adaptive precision or error analysis to ensure reliability.

    Edge Cases and Undefined Behavior

    While \( \arctan\left(\frac{4}{3}\right) \) is well-defined, interactions with singularities or software limitations arise in:
    -

    Visual Representations and Graphical Analysis of arctan(4/3)

    The arctangent function, when evaluated at a specific ratio such as 4/3, reveals deep geometric and analytical insights through visual frameworks. Graphical representations—including unit circle diagrams, functional plots, and dynamic geometric constructions—provide intuitive and rigorous ways to interpret arctan(4/3) in both Euclidean and transcendental contexts. These visual tools bridge abstract algebraic definitions with concrete spatial interpretations, facilitating comprehension of trigonometric relationships, asymptotic behavior, and hypergeometric symmetries.

    Unit Circle Construction of arctan(4/3)

    The angle θ = arctan(4/3) corresponds to the angle subtended by the point (3, 4) on the unit circle, where the adjacent side (x-coordinate) is 3 and the opposite side (y-coordinate) is 4. This construction relies on the right triangle formed by the radius (hypotenuse), the x-axis, and the vertical projection of the point. The hypotenuse, calculated via the Pythagorean theorem, is 5 (since \( \sqrt{3^2 + 4^2} = 5 \)), ensuring the point lies on the unit circle when normalized by dividing by 5.

    Key geometric properties derived from this representation include:

  • Trigonometric ratios:
  • \( \sin(\theta) = \frac{4}{5} \)
  • \( \cos(\theta) = \frac{3}{5} \)
  • \( \tan(\theta) = \frac{4}{3} \) (by definition of arctan).
  • Quadrant placement: The point (3, 4) resides in the first quadrant, confirming θ is an acute angle (\( 0 < \theta < \frac{\pi}{2} \)).
  • Reference angle: The angle’s cosine and sine values align with the 3-4-5 Pythagorean triple, a fundamental example in classical geometry.

    • Plot of the Function y = arctan(x) Near x = 4/3

    The function y = arctan(x) exhibits smooth, bounded behavior across its domain (\( x \in \mathbb{R} \)), with critical features near x = 4/3 ≈ 1.333 that include:
  • Asymptotic behavior:
  • As \( x \to +\infty \), \( y \to \frac{\pi}{2} \) (approaching 1.5708 radians).
  • As \( x \to -\infty \), \( y \to -\frac{\pi}{2} \) (approaching -1.5708 radians).
  • Inflection point: The function’s second derivative, \( y''(x) = \frac{2x(1 - x^2)}{(1 + x^2)^3} \), equals zero at x = 0 and x = ±1. Near x = 4/3, the concavity remains positive (since \( y''(4/3) > 0 \)), indicating a locally convex curve.
  • Slope at x = 4/3: The derivative \( y'(x) = \frac{1}{1 + x^2} \) evaluates to \( \frac{1}{1 + (4/3)^2} = \frac{9}{25} = 0.36 \), representing the tangent line’s gradient at this point.

    • A plot of y = arctan(x) near x = 4/3 would show:

  • A horizontal asymptote at \( y = \frac{\pi}{2} \) (right-hand side).
  • A smooth transition through the origin (0,0) with increasing slope as \( x \) approaches 0 from the positive side.
  • The point (4/3, arctan(4/3)) ≈ (1.333, 0.9273) marked, illustrating the function’s value at this specific input.

    • Trigonometric Value Mapping for arctan(4/3)

    The following table summarizes the primary trigonometric functions evaluated at θ = arctan(4/3), derived from the unit circle construction and right-triangle definitions:
    Function Value Exact Form Decimal Approximation
    sin(θ) 4/5 0.8
    cos(θ) 3/5 0.6
    tan(θ) 4/3 1.333...
    csc(θ) 5/4 1.25
    sec(θ) 5/3 1.666...
    cot(θ) 3/4 0.75
    These values are consistent with the right triangle formed by sides 3, 4, and 5, scaled to the unit circle. The exact forms are rational due to the Pythagorean triple, while decimal approximations provide numerical context for computational applications.

    Dynamic Right Triangle Animation for arctan(4/3)

    A dynamic geometric animation of a right triangle with legs 3 (adjacent) and 4 (opposite) can illustrate θ = arctan(4/3) through interactive transformations:
  • Initial state: The triangle is fixed with the right angle at the origin, the adjacent side along the x-axis, and the opposite side extending vertically.
  • Scaling animation: The triangle is uniformly scaled to fit within the unit circle by dividing all sides by 5, preserving the angle θ while normalizing the hypotenuse to 1.
  • Angle visualization: A rotating arc or sector highlights θ, with labels for the adjacent and opposite sides updating in real-time to reflect their ratios (4/3).
  • Unit circle overlay: The animated triangle is superimposed on the unit circle, demonstrating that the terminal point of θ lies at (3/5, 4/5) after normalization.
  • Parameter control: Sliders adjust the triangle’s side lengths (e.g., scaling 3 and 4 independently) to show how θ changes continuously, reinforcing the inverse relationship between tan(θ) and arctan(x).

    • This animation emphasizes the inverse relationship between the tangent and arctangent functions, where varying the opposite/adjacent ratio directly modulates the angle θ.

    Hyperbolic Reflection of arctan(4/3)

    The arctangent function exhibits a profound connection to hyperbolic geometry through its reflection properties in the unit hyperbola xy = 1. For any real number x > 0, the following identity holds:
    The angle θ = arctan(x) satisfies the hyperbolic reflection property:
    \[ \text{arctan}(x) + \text{arctan}\left(\frac{1}{x}\right) = \frac{\pi}{2} \]
    When applied to x = 4/3, this yields:
    \[ \text{arctan}\left(\frac{4}{3}\right) + \text{arctan}\left(\frac{3}{4}\right) = \frac{\pi}{2} \]
    This symmetry arises from the inverse relationship between x and 1/x on the hyperbola xy = 1, where the tangent of complementary angles in the unit circle corresponds to reciprocal slopes in hyperbolic space.
    The geometric interpretation involves:
  • Complementary angles: The angles arctan(4/3) and arctan(3/4) sum to π/2, mirroring the Pythagorean complementarity of their tangent values.
  • Hyperbolic tangent: The identity extends to the inverse hyperbolic tangent function, artanh(x), which is defined for \( |x| < 1 \) and relates to logarithmic expressions.
  • Advanced applications: This property underpins transformations in complex analysis, particularly in the study of Möbius transformations and conformal mappings, where arctangent functions map lines to circles or hyperbolas.

    Arctan 4 3 exemplifies how a seemingly simple ratio can unlock complex geometric and algebraic principles, from slope gradients in engineering to integration strategies in calculus. Its interplay with complementary angles and numerical methods highlights the interplay between exact values and computational approximations. By mastering this angle, practitioners gain not only a deeper appreciation for trigonometric identities but also practical tools for solving diverse mathematical challenges. Whether through analytical derivations or digital implementations, arctan 4 3 remains a cornerstone of applied mathematics.

    • FAQ

    What is arctan(4/3) expressed in terms of π?

    The exact value of arctan(4/3) cannot be simplified to a basic fraction of π. It is approximately 0.9273 radians (or 53.13°), but no simple π-based expression exists for this ratio.

    How do you convert arctan(4/3) to degrees?

    arctan(4/3) ≈ 53.13010235° when converted from radians to degrees. This is the angle whose tangent is 4/3.

    What is the radian value of arctan(4/3)?

    arctan(4/3) ≈ 0.927295218 radians. This is the exact value in radians for the angle whose tangent is 4/3.

    How do you express arctan(4/3) in degrees?

    arctan(4/3) ≈ 53.13°. This is the angle in degrees where the opposite side is 4 and the adjacent side is 3 in a right triangle.

    What is the value of arctan(4/30)?

    arctan(4/30) ≈ 0.1333 radians (or ≈ 7.602°). This is the angle whose tangent is 4/30 (simplified to 2/15).

    How do you calculate arctan(4/3) × (180/π)?

    Multiplying arctan(4/3) in radians by (180/π) converts it to degrees, yielding ≈ 53.13°. This is the standard conversion from radians to degrees.

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