Problem 1 - Algebra and Trigonometry

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

The formula is \( \sin 2x = 2 \sin x \cos x \).

Step 1: Identify Known Values

The exercise provides the known values: \( \sin x \) and \( \cos x \). These are the trigonometric functions of angle \( x \).

Step 2: Recognize the Formula Required

We need to find \( \sin 2x \), which means we must use the double angle formula for sine. This formula relates \( \sin x \) and \( \cos x \) to \( \sin 2x \).

Step 3: State the Double Angle Formula for Sine

The double angle formula for sine is: \( \sin 2x = 2 \sin x \cos x \). This formula gives us \( \sin 2x \) directly in terms of \( \sin x \) and \( \cos x \).

Key Concepts

Double Angle Formula

The Double Angle Formula is a powerful tool in trigonometry. It allows us to express functions of double angles in terms of single angles. For instance, knowing the values for \( \sin x \) and \( \cos x \), we can compute \( \sin 2x \) and \( \cos 2x \) using specific formulas.

When working with sine, the formula is \( \sin 2x = 2 \sin x \cos x \). This indicates that the sine of a double angle (\( 2x \)) is twice the product of the sine and cosine of the original angle (\( x \)).

• This formula helps us find \( \sin 2x \) without having to measure angle \( 2x \) directly.
• It's especially useful in solving equations or during integration in calculus.

To apply the double angle formula effectively, always ensure you have the values of \( \sin x \) and \( \cos x \). Substituting these values into the formula will yield \( \sin 2x \) immediately.

In summary, the Double Angle Formula for sine is both simple and crucial in simplifying trigonometric expressions and solving real-world problems.

Sine Function

The sine function is one of the fundamental functions in trigonometry, describing how angles relate to the sides of a right triangle. It is commonly written as \( \sin x \), representing the ratio of the length of the side opposite angle \( x \) over the hypotenuse of the triangle.

Key properties of the sine function include:

• Range: The output of \( \sin x \) is always between -1 and 1.
• Periodicity: The sine function is periodic with a period of \( 2\pi \), meaning \( \sin(x + 2\pi) = \sin x \).
• Symmetry: Sine is an odd function. This means \( \sin(-x) = -\sin x \).

Many real-world phenomena involve sinusoidal functions, like sound waves and alternating current in electricity. Hence, understanding the qualities of the sine function allows us to model and predict these phenomena effectively.

When using the sine function with a double angle, like in \( \sin 2x \), the Double Angle Formula offers a swift way to calculate it by linking it back to the primary angle's sine and cosine values.

Cosine Function

The cosine function, written as \( \cos x \), is another primary trigonometric function. It helps in understanding the relationship between an angle and the coordinates of a point on a unit circle. Specifically, \( \cos x \) gives the ratio of the adjacent side's length to the hypotenuse in a right triangle.

Important features of the cosine function include:

• Range: Just like sine, \( \cos x \) values range from -1 to 1.
• Periodicity: The cosine function repeats every \( 2\pi \), implying \( \cos(x + 2\pi) = \cos x \).
• Symmetry: Unlike sine, cosine is an even function, so \( \cos(-x) = \cos x \).

Cosine is frequently used in modeling cyclic phenomena, similar to the sine function. When analyzing angular motion, signals, or any process that involves rotation, cosine plays a critical role.

For computing \( \sin 2x \) using the double angle formula \( \sin 2x = 2 \sin x \cos x \), knowledge of both \( \sin x \) and \( \cos x \) is essential. This illustrates how closely intertwined the sine and cosine functions are in trigonometry.