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Inverse trigonometric functions help us find the angle when the value of a trigonometric function is known. For instance, if \(\cos(\theta) = x\), then the inverse function \(\cos^{-1}(x)\) gives us the angle \(\theta\). The range of \(\cos^{-1}\) is from 0 to \(\pi\) radians. It is important to remember that these functions return angles in specific quadrants to maintain their inverse property.
Using \(\cos^{-1}\) is crucial when solving problems involving angles, especially in trigonometry and geometry. It allows you to convert back from a ratio to an angle, setting the stage for evaluating more complex expressions like double angles or applying coordinate systems.

Double angle formulas are pivotal in trigonometry, as they allow you to express trigonometric functions of twice an angle in terms of functions of the original angle. In the exercise, we used the formula for tangent:

• \( \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \)

These formulas are derived from the addition formulas of trigonometric functions, and they are useful in simplifying complex expressions or solving equations where the variable terms appear as multiples of each other.
The key to applying these formulas effectively is knowing the single angle values first. Once you have \(\tan(\theta)\), the double angle formula allows you to extrapolate to \(\tan(2\theta)\) quickly, saving time and reducing errors in longer calculations.

The Pythagorean theorem is a fundamental principle in geometry that relates the sides of a right triangle. It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides:

• \(c^2 = a^2 + b^2\)

In the solution, we use this theorem to find the length of the opposite side when \(\cos(\theta)\) and one side are known. For example, given \(\cos(\theta) = \frac{1}{4}\) with an adjacent side of 1 and hypotenuse of 4, we solve:

• \(\text{opposite} = \sqrt{4^2 - 1^2} = \sqrt{15}\)

This value then helps in finding \(\tan(\theta)\).
Utilizing the Pythagorean theorem in trigonometry ensures all side lengths and trigonometric ratios we calculate are valid and consistent with basic geometric principles.

The tangent function, \(\tan(\theta)\), is one of the fundamental trigonometric functions and is defined as the ratio of the opposite side to the adjacent side in a right triangle:

• \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)

In our exercise, we calculated \(\tan(\theta)\) using the values we got from the Pythagorean theorem. Knowing \(\tan(\theta)\) allows us to use the double angle formula effectively.
Understanding the behavior of the tangent function is crucial, as it helps reveal insights about angles greater than 90 degrees through periodicity and its relationships with other trigonometric functions. Practical applications of the tangent function range from solving for angles in navigational contexts to simplifying complex trigonometric expressions.