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The inverse tangent, also known as arctangent, is a function that works to find the angle whose tangent is a given number. If we have an expression like \( \tan^{-1}(x) \), it tells us the angle \( \theta \) for which the tangent ratio equals \( x \). This is crucial when working with right triangles as it helps us determine specific angle measures when side lengths are known. In this exercise, \( \tan^{-1} \frac{3}{4} \) indicates that we have an angle whose tangent is \( \frac{3}{4} \). This ratio implies a right triangle where the opposite side is 3 and the adjacent side is 4. It's a handy tool when working with trigonometric identities, and especially when calculator use is restricted. Understanding this concept facilitates translating between angles and side ratios effectively.

A right triangle is a foundational shape in trigonometry, featuring one 90-degree angle. Recognizing how the sides of a right triangle relate to its angles through trigonometric functions is essential.

• One side is known as the hypotenuse, which is the longest side, facing the right angle.
• The other sides are the legs: the opposite side and the adjacent side, relative to a chosen angle.

For the given exercise, our right triangle has one leg of length 3 (opposite) and another of length 4 (adjacent). The relationships of these sides to the angles are key to solving trigonometric problems. With these sides known, other trigonometric functions of the angle, such as cosine or sine, can be calculated using the Pythagorean theorem, which leads us to the next key concept.

The double angle identity for cosine is a formula used to find the cosine of double an angle if we know the initial angle's sine and cosine. It is expressed as:

• \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)

This identity is especially useful for evaluating expressions that involve changing from the original angle to twice that angle. In the exercise at hand, after calculating \( \sin \theta = \frac{3}{5} \) and \( \cos \theta = \frac{4}{5} \), the identity helps us find \( \cos(2\theta) \). By plugging in these values, we derive \( \cos(2\theta) = \frac{16}{25} - \frac{9}{25} = \frac{7}{25} \). Knowing these identities expands our toolkit for manipulating and simplifying trigonometric expressions.

The Pythagorean theorem is a critical mathematical property for analyzing right triangles. It states that the square of a right triangle's hypotenuse is equal to the sum of the squares of the other two sides. Mathematically, it is expressed as:

• \( a^2 + b^2 = c^2 \)

Here, \( a \) and \( b \) are the triangle's legs, while \( c \) is the hypotenuse. In the context of our exercise, using this theorem with the legs 3 and 4 allows us to calculate the hypotenuse as \( \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \). With the hypotenuse known, we can further deduce the sine and cosine of the angle, making it a pivotal step in solving the entire problem. The theorem is a fundamental piece in solving many geometrical and trigonometric challenges.