In Exercises 1–10, use substitution to determine whether the given x-value is a solution of the equation. __ √ 2 𝝅 cos x = ------- , x = ------ 2 4

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1. Substitute the given value of $x=\frac{\pi }{4}$ into the equation $\cos x=\frac{\sqrt{2}}{2}$.
2. Recall the trigonometric identity: $\cos \frac{\pi }{4}=\frac{\sqrt{2}}{2}$.
3. Evaluate $\cos \frac{\pi }{4}$ using the identity to verify if it equals $\frac{\sqrt{2}}{2}$.
4. Since $\cos \frac{\pi }{4}=\frac{\sqrt{2}}{2}$, the given $x$-value satisfies the equation.
5. Conclude that $x=\frac{\pi }{4}$ is indeed a solution to the equation $\cos x=\frac{\sqrt{2}}{2}$.

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function defined as the ratio of the adjacent side to the hypotenuse in a right triangle. It is periodic with a period of 2π and is commonly used in various applications, including wave functions and oscillations. Understanding the properties of the cosine function is essential for solving equations involving angles.

Substitution Method

The substitution method involves replacing a variable in an equation with a specific value to determine if that value satisfies the equation. In this context, substituting the given x-value into the cosine function allows us to check if the left-hand side equals the right-hand side of the equation. This method is a straightforward approach to verify solutions in algebra and trigonometry.

Radian Measure

Radian measure is a way of measuring angles based on the radius of a circle. One radian is the angle formed when the arc length is equal to the radius. In this problem, the x-value is given in radians (π/4), which is crucial for accurately evaluating the cosine function. Understanding how to convert between degrees and radians is vital for solving trigonometric equations.