Problem 36 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the Law of sines or the Law of cosines can be used to find another measure of the triangle. Then solve the triangle. $$a=11, \quad b=13, \quad c=7$$

Step-by-Step Solution

Answer

Calculate the measures of the angles using the steps above, by applying the Law of Cosines first, then the Law of Sines and lastly by subtracting the two calculated angles from 180 degrees to find the third angle.

Step 1: Applying Law of Cosines

Use Law of Cosines to find one of the angles, let's find the angle \(A\) for instance. The formula: \[A = \cos^{-1}\left(\dfrac{b^{2} + c^{2} - a^{2}}{2bc}\right)\] Substitute the lengths \(a=11\), \(b=13\) and \(c=7\), and then calculate.

Step 2: Using Law of Sines

Now that we have one angle, we can use the Law of Sines to find another one. For example, we can find angle \(B\). Then, by using a calculator to solve for \(B\) by the equation \[B = \sin^{-1}\left(\dfrac{b}{c}\sin A\right)\] After that, substitute \(c=7\), \(b=13\) and \(A\) that we calculated in the previous step, and apply the equation.

Step 3: Final Angle Calculation

As we know, the sum of angles in a triangle is 180 degrees. Hence, the last angle \(C\) can be found by subtracting the sum of angles \(A\) and \(B\) from 180. That is, \[C = 180 - A - B\]. Afterward, substitute the values of \(A\) and \(B\), which were found in previous steps to get the value of \(C\).

Key Concepts

Law of Sines

The Law of Sines is an essential tool in trigonometry and is used to find unknown parts of a triangle. It states that the ratio of the length of a side of a triangle to the sine of its opposite angle is the same for all sides of the triangle. This means that for triangle ABC:

• \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

This law is particularly useful when dealing with cases of "ASA" (angle-side-angle) or "AAS" (angle-angle-side). It helps when two angles and one side, or two sides and a non-included angle, are known.
The Law of Sines is used when you do not have a right triangle, and it's helpful when another method, such as the Law of Cosines, does not provide a direct solution. In the provided exercise, once one angle is solved using the Law of Cosines, the Law of Sines becomes applicable to find the other angles.

triangle

A triangle is a three-sided polygon, one of the simplest geometric forms in mathematics. Each triangle consists of three edges and three vertices. It's versatile and appears in various mathematical problems and real-world situations.
Triangles can be classified by their sides:

• Equilateral - all sides equal.
• Isosceles - two sides equal.
• Scalene - all sides different.

They can also be categorized by their angles:

• Acute - all angles less than 90°.
• Right - one angle exactly 90°.
• Obtuse - one angle greater than 90°.

Understanding these basic properties of triangles is crucial when applying trigonometric principles, like those shown in the exercise, to solve for unknown sides or angles.

angle calculation

Calculating angles in a triangle often involves understanding and applying trigonometric laws and properties. In any triangle, the sum of all three interior angles is always 180 degrees.
To calculate a specific angle, you might need to:

• Use the Law of Cosines if you know all three sides and need to find an angle. This can be expressed as:\[ A = \cos^{-1}\left(\dfrac{b^{2} + c^{2} - a^{2}}{2bc}\right) \]
• In case some angles are known, the remaining angles can be calculated directly using:\[ C = 180 - A - B \]

After finding one angle through calculations, as seen in the exercise, other angles can usually be determined more simply using the Law of Sines or by the basic angle sum property.

trigonometry

Trigonometry is a branch of mathematics that deals with the study of triangles, specifically right-angled triangles. However, its principles are also applicable to solving non-right triangles using laws such as the Law of Sines and the Law of Cosines as discussed in your exercise.
Key functions in trigonometry include the sine, cosine, and tangent functions, which relate the angles to the lengths of the sides of a triangle. Here’s a quick rundown:

• Sine (sin): Opposite side over hypotenuse for a right-angled triangle.
• Cosine (cos): Adjacent side over hypotenuse.
• Tangent (tan): Opposite side over adjacent side.

Beyond right-angled triangles, trigonometry uses these concepts to solve any triangle, applying laws to find unknown sides and angles, making it a powerful tool in mathematics and science. Understanding and mastering trigonometry enhances problem-solving skills significantly in both academic and practical applications.