In Exercises 1–10, use substitution to determine whether the given x-value is a solution of the equation. __ √ 3 𝝅sin x = ------- , x = ------- 2 6

Verified step by step guidance

1. Step 1: Identify the given equation and the x-value to substitute. The equation is \( \sin x = \frac{\sqrt{3}}{2} \) and the x-value is \( x = \frac{\pi}{6} \).
2. Step 2: Recall the sine value for common angles. \( \sin \frac{\pi}{6} \) is a well-known trigonometric value.
3. Step 3: Substitute \( x = \frac{\pi}{6} \) into the equation \( \sin x = \frac{\sqrt{3}}{2} \) to check if it satisfies the equation.
4. Step 4: Calculate \( \sin \frac{\pi}{6} \) using the unit circle or trigonometric identities.
5. Step 5: Compare the calculated value of \( \sin \frac{\pi}{6} \) with \( \frac{\sqrt{3}}{2} \) to determine if they are equal, thus verifying if \( x = \frac{\pi}{6} \) is a solution.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to ratios of sides in right triangles. The sine function, for example, is defined as the ratio of the length of the opposite side to the hypotenuse. Understanding these functions is essential for solving equations involving angles and determining their values.

Substitution Method

The substitution method involves replacing a variable in an equation with a specific value to check if it satisfies the equation. In this context, substituting the given x-value into the sine function allows us to verify if the left-hand side equals the right-hand side of the equation. This method is fundamental in solving equations and verifying solutions.

Radians and Degrees

Angles can be measured in degrees or radians, with radians being the standard unit in trigonometry. The given x-value is expressed in radians, which is crucial for accurate calculations in trigonometric functions. Understanding the conversion between degrees and radians is important for correctly interpreting and solving trigonometric equations.