91Ó°ÊÓ

Quadratic equations are an essential part of mathematics, and they often appear in various forms throughout different mathematical problems, including trigonometry. A standard quadratic equation is written as: \[ ax^2 + bx + c = 0 \] In the context of trigonometric equations, we sometimes have variables such as \( \cos \theta \) taking the place of \( x \). For example, we might transform an equation like \( 4 \cos^2 \theta + 5 \cos \theta - 6 = 0 \) into a typical quadratic form to make it easier to solve.The coefficients \( a \), \( b \), and \( c \) represent numerical values that, in this problem, are 4, 5, and -6, respectively.

• The solutions to the quadratic equation can be found using the quadratic formula:

\[ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula gives the potential roots of the quadratic equation, which can then be further analyzed to fit the context such as angles or distances in real problems.

The cosine function is a fundamental trigonometric function that helps relate the angles of a triangle to the lengths of its sides. In a right triangle, the cosine of an angle \( \theta \) is defined as the adjacent side over the hypotenuse.Furthermore, the cosine function has a range from -1 to 1 and is periodic with a period of \( 360^{\circ} \) (or \( 2\pi \) in radians). This period indicates that the cosine function repeats its values every complete circle, which is an important aspect when solving trigonometric equations.In the equation provided, \( \cos \theta \) is framed into a quadratic expression allowing the use of algebraic methods to find the angle solutions. The solutions for \( \cos \theta = u \) within the valid range are the only applicable solutions since \( \cos \theta = -2 \) is outside the permissible range. When solving for \( \theta \) given \( \cos \theta = 0.75 \), using the inverse cosine function or \( \cos^{-1} \) helps us find the angle solutions in degrees, ensuring they fit within the original defined range of the problem.

Solving for angle solutions involves determining the specific angles that satisfy the trigonometric equation within a specified interval. In trigonometry, angle solutions are often expected to be within a certain range, such as between \( 0^{\circ} \) and \( 360^{\circ} \), which corresponds to one full cycle of the unit circle.In this exercise, after establishing that \( \cos \theta = 0.75 \), the angle \( \theta \) can be found using \( \theta = \cos^{-1}(0.75) \). This operation yields an angle of approximately \( 41.41^{\circ} \).

• Because the cosine function is positive in the first and fourth quadrants, these angles would include:
• First quadrant: \( \theta = 41.41^{\circ} \)
• Fourth quadrant: \( \theta = 360^{\circ} - 41.41^{\circ} = 318.59^{\circ} \)

Therefore, these angles are the valid solutions in the specified interval. The double-checking of these solutions ensures completeness and correctness, providing distinct solutions that fall into allowable values for cosine-based angles.