Problem 87 - Precalculus Essentials

Question

In Exercises \(82-89,\) use words to describe the formula for: the sine of half an angle.

Step-by-Step Solution

Answer

The sine of half an angle is equal to the positive or negative square root of half the quantity of one minus the cosine of the original angle.

Step 1: Recall the formula for the sine of half an angle

The formula for the sine of half an angle is described as follows: for any angle \( \theta \), the sine of half that angle is equal to plus or minus the square root of one half of the quantity one minus the cosine of that angle. This is represented as either \( \sin{\frac{\theta}{2}} = \pm \sqrt{\frac{1 - \cos{\theta}}{2}} \). The positive or negative sign is determined by the quadrant in which the half angle falls.

Step 2: Express the formula in words

In words, the above formula can be stated as follows: 'The sine of half an angle is equal to the positive or negative square root of half the quantity of one minus the cosine of the original angle.'

Key Concepts

Sine Function

The sine function is a fundamental aspect of trigonometry that relates to the angles of a right triangle and the ratios of its sides. It is specifically defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. Here is a simple way to understand this:

• If you imagine a right triangle, the sine of an angle \( \theta \) is the length of the side opposite the angle \( \theta \) divided by the length of the hypotenuse.
• In a unit circle, the sine function corresponds to the y-coordinate of a point corresponding to an angle, measured from the positive x-axis.

The sine function is periodic, with a period of \( 2\pi \), meaning that it repeats its values every \( 360\) degrees or \( 2\pi \) radians. This oscillating nature makes it useful in modeling wave patterns and oscillations. Understanding sine is crucial for grasping more complex trigonometric concepts, such as the half-angle formulas.

Half-Angle Formulas

Half-angle formulas are useful trigonometric identities that express trigonometric functions of half angles in terms of the function of the original angle. The half-angle formula for sine is:\[\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}.\]

• This identity allows you to find the sine of half an angle if you know the cosine of the original angle \( \theta \).
• The plus or minus sign depends on the quadrant in which the resulting angle \( \frac{\theta}{2} \) lies.
• This formula is very effective for solving trigonometric problems when you need to find values involving half angles without directly measuring them.

Mastering half-angle formulas can be extremely useful in calculus and physics, where precise values for angles and their corresponding trigonometric functions are required.

Cosine Function

The cosine function, much like sine, is one of the primary functions in trigonometry. It also relates an angle in a right triangle to the lengths of the sides. For cosine:

• The cosine of an angle \( \theta \) is the ratio of the adjacent side to the hypotenuse in a right triangle.
• In the unit circle, cosine corresponds to the x-coordinate of the point on the circle at a given angle from the positive x-axis.

The cosine function is also periodic, with the same period as the sine function, i.e., \( 2\pi \). It helps in determining angles and dimensions in scenarios involving waves and circular motion.
Understanding how cosine integrates into the half-angle formula is crucial. The bare bones of the half-angle formula for sine, outlined earlier, involves the cosine function, highlighting the interconnectivity between these trigonometric functions. By understanding cosine, you are better prepared to work with more complex trigonometric identities.