Problem 34 - Algebra and Trigonometry

Question

Solve each triangle. $$ A=50^{\circ}, B=55^{\circ}, c=9 $$

Step-by-Step Solution

Answer

C = 75°, a ≈ 7.33, b ≈ 7.55

Step 1: Find the Missing Angle

First, use the Triangle Sum Theorem which states that the sum of the angles in a triangle is always 180 degrees. Subtract the given angles from 180 degrees to find the missing angle. \[ C = 180^{\text{°}} - A - B \]\[ C = 180^{\text{°}} - 50^{\text{°}} - 55^{\text{°}} \] \[ C = 75^{\text{°}} \]

Step 2: Use the Law of Sines

Next, use the Law of Sines to find the lengths of sides a and b. The Law of Sines states:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]First, solve for side 'a'. \[ \frac{a}{\sin 50^{\text{°}}} = \frac{9}{\sin 75^{\text{°}}} \]Cross-multiply and solve for 'a':\[ a = \frac{9 \cdot \sin 50^{\text{°}}}{\sin 75^{\text{°}}} \] Compute the values using a calculator: \[ a \approx 7.33 \]

Step 3: Solve for Side 'b'

Now, use the Law of Sines to find side 'b'.\[ \frac{b}{ \sin 55^{\text{°}} } = \frac{9}{ \sin 75^{\text{°}} } \]Cross-multiply and solve for 'b':\[ b = \frac{9 \cdot \sin 55^{\text{°}}}{\sin 75^{\text{°}}} \]Compute the values using a calculator: \[ b \approx 7.55 \]

Key Concepts

Triangle Sum Theorem

The Triangle Sum Theorem states that the sum of the three interior angles of a triangle is always 180 degrees. This theorem is fundamental for solving any triangle, as it helps in determining the unknown angles when the other two angles are known.
In the given problem, we had angles \( A = 50^{\text{°}} \) and \( B = 55^{\text{°}} \). To find angle C, we simply subtract the sum of angles A and B from 180 degrees:
\[ C = 180^{\text{°}} - 50^{\text{°}} - 55^{\text{°}} = 75^{\text{°}} \]
Breaking it down, we found that angle C equaled 75 degrees by applying this theorem.

Angle Sum Property

Closely related to the Triangle Sum Theorem, the angle sum property asserts that the interior angles of a triangle will always sum to 180 degrees.
This property is crucial in geometry and helps us solve for unknown angles within a triangle.
Here’s how we used it in the problem:
We knew two angles (50 degrees and 55 degrees) which summed up to 105 degrees.
By subtracting this from 180 degrees, it led us directly to the third angle, 75 degrees.
After applying this property consistently, it becomes second nature when solving various geometric problems.

Solving Triangles

Solving triangles means finding the unknown sides and angles in a triangle. Typically, we use specific trigonometric rules or theorems to find these unknowns.
In our exercise, we worked with both given angles and side lengths alongside trigonometric functions to arrive at the solution.
To solve given triangles, you can follow these steps:

• Find the missing angle using Triangle Sum Theorem.
• Use trigonometric ratios or the Law of Sines/Law of Cosines to find side lengths.

In our provided problem, we used the Law of Sines to determine the unknown sides once we had all the angles.

Sine Function

The sine function is a fundamental trigonometric function defined for an angle in a right-angled triangle as the ratio of the length of the opposite side to the hypotenuse.
Mathematically, it is expressed as: \[ \text{sin}( \theta ) = \frac{\text{opposite side}}{\text{hypotenuse}} \]
Understanding the sine function is essential when dealing with the Law of Sines, as it hinges directly on this ratio.
In our problem, we used the sine function as follows: \[ \frac{a}{\text{sin} 50^{\text{°}}} = \frac{9}{\text{sin} 75^{\text{°}}} \] and \[ \frac{b}{\text{sin} 55^{\text{°}}} = \frac{9}{\text{sin} 75^{\text{°}}} \]
These expressions rely directly on understanding the function of sine within triangle calculations.

Trigonometric Ratios

Trigonometric ratios are relationships between the sides and angles of a triangle, typically defined using sine, cosine, and tangent functions.
These ratios are utilized to solve for unknown sides or angles within right and non-right triangles.
The main trigonometric ratios include:

• Sine (sin)
• Cosine (cos)
• Tangent (tan)

In our problem, we specifically used the ratio defined by the Law of Sines which compares the lengths of sides to the sines of their opposite angles.
This allowed us to solve for the sides 'a' and 'b'.
Understanding these ratios is key to applying them correctly and solving various triangle-related problems.