In oblique triangle ABC, A = 34°, B = 68°, and a = 4.8. Find b to the nearest tenth.

Verified step by step guidance

1. Step 1: Use the Law of Sines, which states \( \frac{a}{\sin A} = \frac{b}{\sin B} \).
2. Step 2: Substitute the known values into the equation: \( \frac{4.8}{\sin 34^\circ} = \frac{b}{\sin 68^\circ} \).
3. Step 3: Solve for \( b \) by multiplying both sides by \( \sin 68^\circ \): \( b = \frac{4.8 \cdot \sin 68^\circ}{\sin 34^\circ} \).
4. Step 4: Calculate the sine values using a calculator: \( \sin 34^\circ \) and \( \sin 68^\circ \).
5. Step 5: Substitute the sine values back into the equation to find \( b \) to the nearest tenth.

Key Concepts

Law of Sines

The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. This can be expressed as a/sin(A) = b/sin(B) = c/sin(C). This law is particularly useful for solving oblique triangles, where not all angles and sides are known, allowing us to find unknown side lengths or angles.

Oblique Triangle

An oblique triangle is a triangle that does not contain a right angle. It can be either acute (all angles less than 90°) or obtuse (one angle greater than 90°). Solving oblique triangles often requires the use of the Law of Sines or the Law of Cosines, depending on the information given and what needs to be found.

Angle Measures

In trigonometry, angles are typically measured in degrees or radians. In this problem, angles A and B are given in degrees, which is a common unit for measuring angles in triangles. Understanding how to work with angle measures is crucial for applying trigonometric laws effectively, especially when calculating unknown angles or sides.