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

Verified step by step guidance

1. Recognize that the expression involves a double angle identity for cosine: \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \).
2. Identify \( \theta = \arctan(\frac{4}{3}) \), which means \( \tan(\theta) = \frac{4}{3} \).
3. Use the identity \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \) to construct a right triangle with opposite side 4 and adjacent side 3.
4. Calculate the hypotenuse using the Pythagorean theorem: \( \text{hypotenuse} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \).
5. Determine \( \cos(\theta) = \frac{3}{5} \) and \( \sin(\theta) = \frac{4}{5} \), then substitute into the double angle identity: \( \cos(2\theta) = \left(\frac{3}{5}\right)^2 - \left(\frac{4}{5}\right)^2 \).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. Understanding these functions is essential for evaluating expressions involving angles, especially when they are derived from inverse functions like arctan. In this case, we need to evaluate the cosine of an angle that is defined by the arctangent function.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan, allow us to find an angle when given a ratio of sides. For example, arctan(4/3) gives the angle whose tangent is 4/3. This concept is crucial for transforming the expression cos(2 arctan(4/3)) into a more manageable form, as it helps us identify the angle we are working with.

Double Angle Formulas

Double angle formulas are trigonometric identities that express trigonometric functions of double angles in terms of single angles. For cosine, the formula is cos(2θ) = cos²(θ) - sin²(θ) or alternatively, cos(2θ) = 2cos²(θ) - 1. Applying this formula will be necessary to evaluate cos(2 arctan(4/3)) effectively, as it allows us to relate the cosine of a double angle to the cosine and sine of the original angle.