Problem 98 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Law of sines or the Law of cosines to solve the triangle. $$A=56^{\circ}, C=38^{\circ}, c=12$$

Step-by-Step Solution

Answer

The unknown angles and sides are \(B=86^{\circ}\), side \(a\) calculated by using the law of sines is approximately equal to 14.71, and side \(b\) calculated also by using the law of sines is approximately equal to 19.79.

Step 1: Determine Angle B

Compute the third angle as \[B = 180^{\circ} - (A + C)\]. Substituting values, \[B = 180^{\circ} - (56^{\circ} + 38^{\circ}) = 86^{\circ}\].

Step 2: Calculate Side a using Law of Sines

The law of sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides. Thus, \[a/sin(A) = c/sin(C)\]. Rearranging, we get \[a = c * sin(A)/sin(C) = 12 * sin(56^{\circ})/sin(38^{\circ})\].

Step 3: Calculate Side b using Law of Sines

Similarly, using the law of sines for angle B, we get \[b = c * sin(B)/sin(C) = 12 * sin(86^{\circ})/sin(38^{\circ})\]. It's essential to use a calculator for the sine calculations to obtain the most accurate results.

Key Concepts

Law of Cosines

The Law of Cosines is an important tool for solving triangles, especially those that are not right-angled. It allows us to find a triangle's angles and sides when we know two sides and the included angle, or when we know all three sides. The Law of Cosines is expressed as:\[c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\]Where:

• \(c\) is the side opposite angle \(C\)
• \(a\) and \(b\) are the other two sides
• \(C\) is the angle opposite side \(c\)

This formula can be rearranged to solve for any of the sides or the included angle when needed.
If you're trying to find an unknown side, substitute the known values into the equation and solve. When finding an angle with known side lengths, rearrange the equation to make \(\cos(C)\) the subject then take the inverse cosine to find the angle. This makes the Law of Cosines versatile and practical for various triangle problems. Understanding and practicing this Law can greatly improve your problem-solving skills in trigonometry.

Triangle Angles

Understanding angles is fundamental to solving triangles. A triangle is defined by three angles and the sum of these angles is always \(180^{\circ}\). This property is key when calculating unknown angles. In the given problem, two angles were provided: \(A = 56^{\circ}\), and \(C = 38^{\circ}\). The third angle \(B\) was found using the relation:\[B = 180^{\circ} - (A + C)\]After substituting, we found \(B = 86^{\circ}\).
Here's a quick summary:

• The sum of the angles in any triangle is \(180^{\circ}\).
• To find an unknown angle, subtract the sum of the known angles from \(180^{\circ}\).

These principles help ensure accurate solutions in trigonometry problems involving angles.
Keeping track of each angle and its relationship to the sides using laws like the Law of Sines ensures that the triangle properties remain consistent and complete.

Trigonometric Ratios

Trigonometric ratios are essential for connecting angles and sides in triangles. The primary trigonometric ratios are sine, cosine, and tangent. In the context of solving triangles, the sine function is particularly useful, especially when applying the Law of Sines.
Here is how it applies:

• \(\sin(\theta) = \frac{{\text{opposite side}}}{{\text{hypotenuse}}}\) in a right triangle.
• In non-right triangles, the Law of Sines states: \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\).

For instance, to find side \(a\), you can rearrange and compute it using:\[a = c \cdot \frac{\sin(A)}{\sin(C)}\]Using trigonometric ratios requires familiarity with a calculator that can compute sine values, typically given in degrees. These ratios provide a bridge between the algebraic and geometric worlds, enabling you to solve a variety of practical and theoretical problems related to triangles.