Problem 34 - 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. $$a=10, \quad b=12, \quad C=70^{\circ}$$

Step-by-Step Solution

Answer

The solution involves first applying the Law of Cosines to find the third side c, and then using the Law of Sines to find angle A. Lastly, the angle sum property of triangles is used to find angle B. The final solution will provide the measures of all sides and angles in the triangle.

Step 1: Apply the Law of Cosines to Find Side c

Using the formula \( c^2 = a^2 + b^2 - 2ab \cos(C) \), substitute the values: \( c^2 = 10^2 + 12^2 - 2 \cdot 10 \cdot 12 \cos(70^{\circ}) \) \n. Solve to find the length of side c.

Step 2: Use the Law of Sines to Find Angle A

After finding c, use the Law of Sines formula \( \frac{a}{\sin(A)} = \frac{c}{\sin(C)} \) . Substitute the known values: \( \frac{10}{\sin(A)} = \frac{c}{\sin(70^{\circ})} \). Solve to find \(\sin(A)\), then use the inverse sin function to find angle A in degrees.

Step 3: Find Angle B Using the Angle Sum Property

Once angles A and C are known, use the angle sum property of triangles (sum of angles in a triangle is 180 degrees) to find angle B. So, \( B = 180^{\circ} - A - C \)

Key Concepts

Law of Cosines

The Law of Cosines is an essential tool in trigonometry for solving triangles, particularly when we know two sides and the included angle, or all three sides of a triangle. This law is a direct extension of the Pythagorean theorem, but it applies to any triangle, not just right-angled ones.
To use the Law of Cosines, we apply the formula:

• \( c^2 = a^2 + b^2 - 2ab \cos(C) \)

Here, \( c \) is the side opposite the angle \( C \), and \( a \) and \( b \) are the other two sides of the triangle.
In our exercise, we used this formula to find side \( c \) when the sides \( a = 10 \), \( b = 12 \), and the angle \( C = 70^\circ \) were known. Simply substitute these values into the equation and solve for \( c \). This will give the length of the side opposite to the given angle, providing a deeper understanding of the triangle's dimensions.

Law of Sines

The Law of Sines is another crucial formula in trigonometry used to solve triangles. This law is especially useful when dealing with non-right triangles. The Law of Sines relates the ratio of each side of a triangle to the sine of its opposite angle. It is expressed as:

• \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \)

When you know at least one angle and its opposite side, along with another side and its opposite angle, you can apply this law.
In our example, after finding side \( c \) using the Law of Cosines, we used the Law of Sines to find angle \( A \). By substituting \( a = 10 \) and \( \sin(70^\circ) \), we solved the equation \( \frac{10}{\sin(A)} = \frac{c}{\sin(70^\circ)} \) to find \( \sin(A) \) and subsequently calculated \( A \) using the inverse sine function.
This approach illustrates how the Law of Sines can solve triangles by finding missing angles once a specific side and angle are known.

Angle Sum Property

The angle sum property is a fundamental principle in geometry. This property states that the sum of the interior angles of any triangle is always \( 180^\circ \). This rule helps us to solve for missing angles once we know at least two angles in the triangle.
In our exercise, once the angles \( A \) and \( C \) were known, finding angle \( B \) was straightforward. We used the angle sum property, expressed as:

• \( B = 180^\circ - A - C \)

By substituting the known angles, we can easily determine the value of angle \( B \).
This property is not only a valuable technique for solving triangles but also reinforces the relationship and constraints within the structure of a triangle.

Solving Triangles

Solving triangles involves finding unknown sides or angles using known values. This requires a combination of various trigonometric laws and properties, depending on the given data. The process typically involves:

• Identifying which laws or properties to use based on the known and unknown elements.
• Applying the Law of Cosines or the Law of Sines when appropriate.
• Using the angle sum property to find missing angles.

For example, given \( a = 10 \), \( b = 12 \), and \( C = 70^\circ \), we first used the Law of Cosines to determine side \( c \).
Then, applying the Law of Sines helped us find angle \( A \). Finally, the angle sum property allowed us to calculate angle \( B \). These steps together equipped us to solve the entire triangle, showcasing how these concepts work harmoniously to find unknown measurements in a triangle.