Solve each triangle ABC.

C = 79° 18', c = 39.81 mm, A = 32° 57'

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Convert the given angles from degrees and minutes to decimal degrees for easier calculation. For example, convert 79° 18' to decimal degrees using the formula: \(\text{decimal degrees} = \text{degrees} + \frac{\text{minutes}}{60}\).

Calculate angle B using the fact that the sum of angles in a triangle is 180°. Use the formula: \(B = 180^\circ - A - C\).

Use the Law of Sines to find the sides a and b. The Law of Sines states: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\).

Rearrange the Law of Sines to solve for side a: \(a = c \times \frac{\sin A}{\sin C}\), and similarly for side b: \(b = c \times \frac{\sin B}{\sin C}\).

Calculate the lengths of sides a and b using the values of angles A, B, C and side c, completing the solution of triangle ABC.

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Triangle Angle Sum Property

The sum of the interior angles in any triangle is always 180°. Knowing two angles allows you to find the third by subtracting their sum from 180°, which is essential for solving the triangle completely.

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4:47

Sum and Difference of Tangent

Law of Sines

The Law of Sines relates the sides and angles of a triangle: (a/sin A) = (b/sin B) = (c/sin C). It is used to find unknown sides or angles when given a combination of angles and sides, such as in this problem.

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Intro to Law of Sines

Angle and Side Notation in Triangles

In triangle notation, angles are labeled with capital letters (A, B, C) and their opposite sides with corresponding lowercase letters (a, b, c). Understanding this correspondence is crucial for applying trigonometric laws correctly.

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