Question

Fill in the blank to make a true statement. \(\sin ^{2} x+\) _______ \(=1\)

Short answer

The missing term is \(\cos^2 x\).

Step-by-step solution

Understand the Pythagorean Identity

The expression given is based on the Pythagorean identity in trigonometry. This identity states that for any angle \(x\), the square of the sine function plus the square of the cosine function equals 1. The identity can be written as: \( \sin^2 x + \cos^2 x = 1 \).

Identify the Missing Component

The problem asks to fill in the blank in the expression \(\sin^2 x + \underline{\quad} = 1\). By comparing this with the Pythagorean identity \( \sin^2 x + \cos^2 x = 1\), we can see that the missing component is \( \cos^2 x \).

Complete the Expression

Fill in the blank with \( \cos^2 x \) to complete the expression. Thus, the full expression is \( \sin^2 x + \cos^2 x = 1 \), which is a true statement based on the standard trigonometric identity.

Key concepts

Trigonometric Identities

Trigonometric identities are equations that hold true for all angles. They are crucial tools in understanding and solving problems in trigonometry. One of the most well-known trigonometric identities is the Pythagorean Identity. This identity relates the square of the sine function and the square of the cosine function for any angle:\[ \sin^2 x + \cos^2 x = 1 \]This equation is true for any angle \( x \), and it reflects a deep relationship between sine and cosine. Trigonometric identities like this one are used to simplify expressions and solve trigonometric equations. To effectively use these identities, it's essential to recognize and apply them appropriately. These identities help us transform complex trigonometric expressions into simpler, more manageable forms.

Sine and Cosine Relationship

The sine and cosine functions are fundamental in trigonometry. They describe the relationship between the angles and side lengths in a right-angled triangle. These functions also have a unique relationship with each other, as evident in the Pythagorean Identity. The square of the sine of an angle plus the square of the cosine of the same angle equates to 1. This highlights how interconnected these two functions are.

Key points about their relationship include:

• The maximum value of both sine and cosine is 1, and the minimum is -1.
• Sine and cosine are periodic functions, repeating every \( 360^\circ \) or \( 2\pi \) radians.
• When graphed, they produce wave-like patterns that mirror each other with a phase difference.

Understanding these properties helps in solving trigonometric problems and analyzing periodic phenomena.

Basic Trigonometry

Basic trigonometry revolves around the study of angles, triangles, and the functions derived from them. It is the foundation for understanding more complex mathematical concepts. In a right-angled triangle, the primary trigonometric functions – sine, cosine, and tangent – are defined as follows:

• Sine (\( \sin \)) of an angle \( x \) is the ratio of the opposite side to the hypotenuse.
• Cosine (\( \cos \)) of an angle \( x \) is the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \)) is the ratio of the sine of an angle to the cosine of the same angle.

These functions are essential for calculating unknown side lengths and angles in triangles. Moreover, basic trigonometry provides the tools needed to understand trigonometric identities, solve equations, and analyze waves and oscillations. Mastery of these basic concepts is crucial for progressing in mathematics and physics.