Find each angle B. Do not use a calculator.


<IMAGE>

Verified step by step guidance

1. Identify the trigonometric function or relationship given in the problem. For example, if the problem involves a right triangle, consider using sine, cosine, or tangent ratios.
2. Express the given information in terms of a trigonometric equation. For instance, if you know the opposite and adjacent sides of a right triangle, you can use \( \tan(B) = \frac{\text{opposite}}{\text{adjacent}} \).
3. Use known trigonometric values or identities to solve for the angle. For example, if \( \tan(B) = 1 \), then \( B = 45^\circ \) because \( \tan(45^\circ) = 1 \).
4. If the problem involves special angles, recall the common angle values and their trigonometric ratios, such as \( 30^\circ, 45^\circ, \) and \( 60^\circ \).
5. Verify your solution by checking if the calculated angle satisfies the original trigonometric equation or relationship.

Key Concepts

Trigonometric Ratios

Trigonometric ratios are relationships between the angles and sides of a right triangle. The primary ratios are sine, cosine, and tangent, defined as the ratios of the lengths of the sides opposite, adjacent, and hypotenuse to the angle in question. Understanding these ratios is essential for solving problems involving angles and side lengths in triangles.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsine, arccosine, and arctangent, are used to find angles when the values of the trigonometric ratios are known. These functions allow us to determine the angle corresponding to a given sine, cosine, or tangent value, which is crucial for solving for angles in trigonometric problems.

Angle Relationships in Triangles

In any triangle, the sum of the interior angles is always 180 degrees. This fundamental property allows us to find unknown angles when some angles are known. Additionally, in right triangles, the relationships between the angles can be explored using complementary angles, where the two non-right angles add up to 90 degrees, aiding in angle calculations.