Problem 56 - Algebra and Trigonometry

Question

Write the expression in terms of sine only. $$ \sin x+\cos x $$

Step-by-Step Solution

Answer

\( \sin x + \sqrt{1 - \sin^2 x} \)

Step 1: Use the Pythagorean identity

We know that \( \sin^2 x + \cos^2 x = 1 \). This identity can help us express \( \cos x \) in terms of \( \sin x \).

Step 2: Express \( \cos x \) in terms of \( \sin x \)

From the identity \( \sin^2 x + \cos^2 x = 1 \), we can deduce that \( \cos^2 x = 1 - \sin^2 x \). Taking the square root gives \( \cos x = \sqrt{1 - \sin^2 x} \) or \( \cos x = -\sqrt{1 - \sin^2 x} \).

Step 3: Substitute \( \cos x \) in the original expression

Replace \( \cos x \) with \( \sqrt{1 - \sin^2 x} \) in the original expression to get \( \sin x + \sqrt{1 - \sin^2 x} \). If the angle \( x \) belongs to the first or fourth quadrants, \( \cos x = \sqrt{1 - \sin^2 x} \) is generally considered.

Key Concepts

Sine Function

The sine function is one of the fundamental building blocks in trigonometry, symbolized as \( \sin \theta \), where \( \theta \) is the angle. It represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. This definition makes the sine function very useful for solving various problems in geometry and physics.

It ranges from -1 to 1, oscillating smoothly, and repeats every \( 2\pi \) radians (360 degrees). Knowing the sine of specific angles can greatly aid in solving trigonometric problems. For example:

• \( \sin(0) = 0 \)
• \( \sin(\frac{\pi}{2}) = 1 \)
• \( \sin(\pi) = 0 \)
• \( \sin(\frac{3\pi}{2}) = -1 \)
• \( \sin(2\pi) = 0 \)

Using the sine function in conjunction with other trigonometric identities allows us to express different angles and lengths, as is done with the Pythagorean identity to express \( \cos x \) in terms of \( \sin x \).

Cosine Function

The cosine function, denoted as \( \cos \theta \), complements the sine function, representing the ratio of the length of the adjacent side to the hypotenuse in a right triangle. Similar to sine, the cosine function offers a range from -1 to 1 and also exhibits periodic waves that repeat every \( 2\pi \) radians.

Some key cosine values include:

• \( \cos(0) = 1 \)
• \( \cos(\frac{\pi}{2}) = 0 \)
• \( \cos(\pi) = -1 \)
• \( \cos(\frac{3\pi}{2}) = 0 \)
• \( \cos(2\pi) = 1 \)

In the exercise, we aimed to express an algebraic expression in terms of sine only, utilizing the Pythagorean identity to rewrite \( \cos x \) as \( \sqrt{1-\sin^2 x} \) or \(-\sqrt{1-\sin^2 x} \). This transformation is essential when the goal is expressing trigonometric expressions in terms of one primary function.

Pythagorean Identity

The Pythagorean identity is a crucial tool in trigonometry. It states that \( \sin^2 x + \cos^2 x = 1 \). This relationship stems from the Pythagorean theorem and is fundamental for deriving various trigonometric identities.

This particular identity offers a way to interconvert sine and cosine functions. For instance, you can express \( \cos^2 x \) in terms of \( \sin^2 x \) by rearranging the identity to \( \cos^2 x = 1 - \sin^2 x \). From here, finding \( \cos x \) involves taking the square root, resulting in either \( \sqrt{1-\sin^2 x} \) or \(-\sqrt{1-\sin^2 x} \).

The sign choice depends on the angle's quadrant positioning. In the context of the exercise, this expression transformation assists in rewriting functions like \( \sin x + \cos x \) solely in terms of sine, facilitating simpler evaluations or integrations when solving mathematical problems. This identity is fundamental for simplifying expressions or solving equations involving trigonometric functions.