Problem 27 - Precalculus Mathematics for Calculus

Question

Use the Law of sines to solve for all possible triangles that satisfy the given conditions. $$a=26, \quad c=15, \quad \angle C=29^{\circ}$$

Step-by-Step Solution

Answer

There are two possible triangles: (1) \( A = 56.2^{\circ}, B = 94.8^{\circ}, C = 29^{\circ} \); (2) \( A = 123.8^{\circ}, B = 27.2^{\circ}, C = 29^{\circ} \).

Step 1: Identify the Given Information

We are given that side \( a = 26 \), side \( c = 15 \), and angle \( C = 29^{\circ} \). The goal is to find all possible triangles using the Law of Sines.

Step 2: Use the Law of Sines to Find \( A \)

The Law of Sines states \( \frac{a}{\sin A} = \frac{c}{\sin C} \). We can use this formula to find \( \sin A \):\[ \frac{26}{\sin A} = \frac{15}{\sin 29^{\circ}} \].Simplifying, we get:\[ \sin A = \frac{26 \cdot \sin 29^{\circ}}{15} \].This calculation results in \( \sin A \approx 0.8374 \).

Step 3: Determine Angle \( A \)

Using the inverse sine function, \( A = \sin^{-1}(0.8374) \approx 56.2^{\circ} \). Since sine is positive in two quadrants, \( A \) could also be \( 180^{\circ} - 56.2^{\circ} = 123.8^{\circ} \). Thus, there are two possible triangles.

Step 4: Calculate Angle \( B \) for Each Triangle

For the first triangle where \( A = 56.2^{\circ} \), angle \( B = 180^{\circ} - A - C = 94.8^{\circ} \).For the second triangle where \( A = 123.8^{\circ} \), angle \( B = 180^{\circ} - A - C = 27.2^{\circ} \).

Step 5: Verify Possible Triangles

Check that these angles form valid triangles. 1. First triangle: \( A = 56.2^{\circ}, B = 94.8^{\circ}, C = 29^{\circ} \). 2. Second triangle: \( A = 123.8^{\circ}, B = 27.2^{\circ}, C = 29^{\circ} \).Both triangles satisfy the triangle inequality and the sum of angles equals \( 180^{\circ} \).

Step 6: Conclusion

There are two possible triangles satisfying the given conditions:1. Triangle with angles \( A = 56.2^{\circ}, B = 94.8^{\circ}, C = 29^{\circ} \).2. Triangle with angles \( A = 123.8^{\circ}, B = 27.2^{\circ}, C = 29^{\circ} \).

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies relationships between the sides and angles of triangles. It is especially powerful for solving problems involving triangles and circles. One of its primary tools is trigonometric functions like sine, cosine, and tangent.

For example, in the Law of Sines, which is part of trigonometry, we relate the lengths of sides of a triangle with the sine of its opposite angles:

• The formula goes \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \).
• Each side of a triangle corresponds to and can be calculated using the sine of its opposite angle.

Trigonometry provides the framework for solving triangles by setting the rules and relationships that all valid triangles must follow.

Triangle Solving

Triangle solving is all about determining unknown elements, such as angles and side lengths, given some known values. This process can be navigated through methods like the Law of Sines or the Law of Cosines.

In solving our specific problem, we first determined side lengths and one angle. Then, by applying the Law of Sines:

• We found a possible angle using the known values.
• Depending on given data, multiple triangles can be solved.

All calculations are rooted in the idea that the sum of the angles in a triangle is always 180 degrees.

Angle Calculations

Calculating angles within triangles is crucial for determining triangle properties. Once any two angle measures are found, the third angle can be determined as the total must equal 180 degrees.

Using the Law of Sines, you can find an angle in the following way:

• You isolate and compute \( \sin A \).
• Apply the inverse sine function to solve for angle \( A \).

In our exercise, we calculated possible values for angle \( A \), choosing solutions that kept all angles meaningful and valid.

Triangle Inequality

The triangle inequality theorem states that the sum of the lengths of any two sides must be greater than the length of the remaining side. This is fundamental in deciding whether a set of given lengths and angles can form a triangle.

When solving triangles, you check:

• Each calculated angle and side must comply with this rule.
• If the sum of two side lengths equals or is less than the third side's length, then such a triangle cannot exist.

In our solution, checking against these constraints ensured that both potential triangles are legitimate.

Inverse Sine Function

The inverse sine function (\( \sin^{-1} \) or arcsin) is the tool used to find angles from known sine values. It is crucial when using the Law of Sines as it helps determine angle measures from partial data.

Here's how you can apply it:

• Calculate a sine value from known sides and angles.
• Use \( \sin^{-1} \) to find the corresponding angle that produces that sine value.

Remember that because the sine is positive in two quadrants of the unit circle, you should consider two possible angles unless restricted by other conditions.