Problem 1 - Precalculus Mathematics for Calculus

Question

If we know the values of \(\sin x\) and \(\cos x,\) we can find the value of \(\sin 2 x\) by using the _____. Formula for sine. State the formula: \(\sin 2 x=\) ______.

Step-by-Step Solution

Answer

Formula: \(\sin 2x = 2 \sin x \cos x\).

Step 1: Identify the Formula for Double Angle Sine

When given the values of \(\sin x\) and \(\cos x\), we can find the value of \(\sin 2x\) using the double angle formula for sine. The formula is:\[sin 2x = 2 \sin x \cos x\]

Step 2: Apply the Formula

To find \(\sin 2x\), simply substitute the known values of \(\sin x\) and \(\cos x\) into the formula \(\sin 2x = 2 \sin x \cos x\) and perform the multiplication to arrive at the answer.

Key Concepts

Double Angle Formula

The double angle formulas are essential tools in trigonometry. They allow you to express trigonometric functions of double angles, like \(2x\), in terms of the functions of a single angle \(x\). The double angle formula for sine is particularly handy when you need to find the sine of an angle that is twice as large as a given one.

For sine, the double angle formula is

• \( \sin 2x = 2 \sin x \cos x \)

To use it, you'll need to know the values of \(\sin x\) and \(\cos x\). Then, by plugging these values into the formula, you can easily find \(\sin 2x\). It's a fantastic shortcut!

This formula is derived from the sum formulas for trigonometric functions, specifically the sine addition formula \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). By setting \(a = b = x\), we simplify it to easily compute complex angles.

Sine Function

The sine function is a fundamental part of trigonometry. It primarily outputs the ratio of the length of the side of a right triangle opposite a given angle to the length of the hypotenuse. This finds wide applications in various fields including physics, engineering, and computer science.

• Expressed mathematically as \( \sin x \), where \(x\) is the angle in question.

The sine of an angle is periodic, meaning it repeats its values in a cyclic manner, specifically, every \(360^\circ\) or \(2\pi\) radians.

Its range values are from -1 to 1, which means, no matter the angle, the sine will never be more than 1 or less than -1.

When incorporating the sine function into a double angle problem, think of it as determining the size of one component of a bigger picture. For double angles, the sine function helps to bridge the smaller angles to larger ones efficiently.

Cosine Function

The cosine function is another critical trigonometric identity that complements the sine function. In a right triangle, the cosine of an angle \(x\) is the ratio of the length of the adjacent side to the hypotenuse.

• Symbolically, this can be written as \( \cos x \).

Much like the sine function, \(\cos x\) is periodic with a cycle of \(360^\circ\) or \(2\pi\) radians, and its range is also between -1 and 1.

The cosine function is useful in calculating not just lengths and angles, but also in resolving components of forces and in waves and oscillations in physics.

In the double angle formula for sine, the role of cosine is significant because both sine and cosine values at a specific angle are needed. Together, they convert simple angle trigonometric values into double angle measurements. This interplay between sine and cosine is a great example of how trigonometric functions work together to solve complex geometry challenges.