Evaluate each expression without using a calculator.
cos (2 arctan (4/3))

Guida verificata passo dopo passo

Recognize that the expression involves a double angle cosine function with an angle defined as \(\theta = \arctan\left(\frac{4}{3}\right)\). So the expression is \(\cos(2\theta)\) where \(\theta = \arctan\left(\frac{4}{3}\right)\).

Recall the double angle identity for cosine: \(\cos(2\theta) = \frac{1 - \tan^2(\theta)}{1 + \tan^2(\theta)}\). This identity is useful because we know \(\tan(\theta)\) from the problem.

Substitute \(\tan(\theta) = \frac{4}{3}\) into the double angle formula: \(\cos(2\theta) = \frac{1 - \left(\frac{4}{3}\right)^2}{1 + \left(\frac{4}{3}\right)^2}\).

Simplify the numerator and denominator separately by squaring \(\frac{4}{3}\) and then performing the subtraction and addition inside the fraction.

After simplification, write the resulting fraction as the value of \(\cos(2\arctan(4/3))\). This completes the evaluation without using a calculator.

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Inverse Trigonometric Functions (Arctan)

The arctan function, or inverse tangent, returns an angle whose tangent is a given number. For example, arctan(4/3) gives an angle θ such that tan(θ) = 4/3. Understanding this allows us to express trigonometric expressions involving arctan in terms of right triangle ratios.

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Introduction to Inverse Trig Functions

Double-Angle Formula for Cosine

The double-angle formula for cosine states that cos(2θ) = cos²θ - sin²θ, or equivalently cos(2θ) = 1 - 2sin²θ or 2cos²θ - 1. This formula helps simplify expressions involving twice an angle, such as cos(2 arctan(x)), by relating it to sine and cosine of the original angle.

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Double Angle Identities

Right Triangle Trigonometry and Ratio Conversion

By interpreting arctan(4/3) as an angle in a right triangle with opposite side 4 and adjacent side 3, we can find the hypotenuse and then determine sine and cosine values. This geometric approach allows us to rewrite trigonometric expressions in exact fractional form without a calculator.

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