Solve each equation for exact solutions.
arcsin x = arctan 3/4

Verified step by step guidance

1. Recognize that the equation is \( \arcsin x = \arctan \frac{3}{4} \), which means the angle whose sine is \( x \) is equal to the angle whose tangent is \( \frac{3}{4} \).
2. Let \( \theta = \arctan \frac{3}{4} \). This means \( \tan \theta = \frac{3}{4} \). We want to find \( x = \sin \theta \).
3. Use the right triangle definition of tangent: if \( \tan \theta = \frac{3}{4} \), then the opposite side is 3 and the adjacent side is 4. Calculate the hypotenuse using the Pythagorean theorem: \( \text{hypotenuse} = \sqrt{3^2 + 4^2} \).
4. Find \( \sin \theta \) using the triangle sides: \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{\sqrt{3^2 + 4^2}} \).
5. Since \( \arcsin x = \theta \), the exact solution for \( x \) is \( \sin \theta \) as found above.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin and arctan, return the angle whose sine or tangent is a given value. Understanding their domains and ranges is essential to find exact angle measures and interpret solutions correctly.

Relationship Between Sine and Tangent

Sine and tangent are related through the sides of a right triangle: tangent equals sine divided by cosine. Knowing this relationship helps convert between arcsin and arctan expressions and find exact values of x.

Exact Values and Right Triangle Ratios

Exact trigonometric values often come from special right triangles or ratios of integer sides. Recognizing the 3-4-5 triangle allows precise calculation of sine and tangent values without approximations.