Which one of the following equations has solution 3π/4
a. arctan 1 = x
b. arcsin √2/2 = x
c. arccos (―√2 /2) = x

Verified step by step guidance

1. Step 1: Understand the problem. We need to determine which inverse trigonometric function equation has a solution of \( \frac{3\pi}{4} \).
2. Step 2: Analyze option (a): \( \arctan(1) = x \). The \( \arctan \) function returns the angle whose tangent is 1. Recall that \( \tan(\frac{\pi}{4}) = 1 \), so \( \arctan(1) = \frac{\pi}{4} \). This does not match \( \frac{3\pi}{4} \).
3. Step 3: Analyze option (b): \( \arcsin(\frac{\sqrt{2}}{2}) = x \). The \( \arcsin \) function returns the angle whose sine is \( \frac{\sqrt{2}}{2} \). Recall that \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \), so \( \arcsin(\frac{\sqrt{2}}{2}) = \frac{\pi}{4} \). This does not match \( \frac{3\pi}{4} \).
4. Step 4: Analyze option (c): \( \arccos(-\frac{\sqrt{2}}{2}) = x \). The \( \arccos \) function returns the angle whose cosine is \(-\frac{\sqrt{2}}{2} \). Recall that \( \cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} \), so \( \arccos(-\frac{\sqrt{2}}{2}) = \frac{3\pi}{4} \).
5. Step 5: Conclude that option (c) is the correct equation since it matches the given solution \( \frac{3\pi}{4} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan, arcsin, and arccos, are used to find angles when given a ratio of sides in a right triangle. For example, arctan(1) gives the angle whose tangent is 1, which corresponds to π/4 radians. Understanding these functions is crucial for solving equations that involve finding angles from given trigonometric values.

Unit Circle

The unit circle is a fundamental concept in trigonometry that defines the relationship between angles and coordinates in a circular context. Each angle corresponds to a point on the circle, where the x-coordinate represents the cosine and the y-coordinate represents the sine of the angle. Knowing the coordinates of key angles, such as 3π/4, helps in determining the values of trigonometric functions and their inverses.

Trigonometric Values at Key Angles

Certain angles, like π/4, π/3, and π/6, have well-known sine, cosine, and tangent values. For instance, sin(3π/4) equals √2/2 and cos(3π/4) equals -√2/2. Recognizing these values allows for quick identification of the correct inverse function that yields a specific angle, which is essential for solving the given equations.