Question

Solve each triangle. $$ A=110^{\circ}, \quad C=30^{\circ}, \quad c=3 $$

Short answer

B = 40°, a ≈ 5.64, b ≈ 3.86

Step-by-step solution

Find the Missing Angle B

The sum of the angles in a triangle is always 180 degrees. Use the formula: \[ B = 180^\text{°} - A - C \] Substitute the given values: \[ B = 180^\text{°} - 110^\text{°} - 30^\text{°} \]Calculate the result: \[ B = 40^\text{°} \]

Use the Law of Sines to Find Side a

Use the Law of Sines formula: \[ \frac{a}{\text{sin}(A)} = \frac{c}{\text{sin}(C)} \] Rearranging to solve for a: \[ a = c \left( \frac{\text{sin}(A)}{\text{sin}(C)} \right) \] Substitute the given values: \[ a = 3 \left( \frac{\text{sin}(110^\text{°})}{\text{sin}(30^\text{°})} \right) \]Calculate the result: \[ a \approx 5.64 \]

Use the Law of Sines to Find Side b

Again, use the Law of Sines formula: \[ \frac{b}{\text{sin}(B)} = \frac{c}{\text{sin}(C)} \] Rearranging to solve for b: \[ b = c \left( \frac{\text{sin}(B)}{\text{sin}(C)} \right) \] Substitute the given values: \[ b = 3 \left( \frac{\text{sin}(40^\text{°})}{\text{sin}(30^\text{°})} \right) \]Calculate the result: \[ b \approx 3.86 \]

Key concepts

Law of Sines

The Law of Sines is a helpful theorem for solving triangles, especially when you know one side length and the opposite angle and seek another side length or angle in the triangle. The formula is: \ \frac{a}{\text{sin}(A)} = \frac{b}{\text{sin}(B)} = \frac{c}{\text{sin}(C)}, \ where \( a, b, \text{ and } c \) are the side lengths, and \( A, B, \text{ and } C \) are the respective opposite angles.

For the given triangle with angles \( A=110^\text{°}\) and \( C=30^\text{°} \), and side \( c=3 \):

• Calculate side \( a \)
\ a = c \frac{\text{sin}(A)}{\text{sin}(C)} = 3 \frac{\text{sin}(110^\text{°})}{\text{sin}(30^\text{°})} \
• Calculate side \( b \)
\ b = c \frac{\text{sin}(B)}{\text{sin}(C)} = 3 \frac{\text{sin}(40^\text{°})}{\text{sin}(30^\text{°})} \ This method, using proportions involving angles and the opposite sides, allows you to efficiently and accurately solve for the unknown sides.

Always remember that you need at least one angle-side pair to use the Law of Sines. It becomes a powerful tool for non-right triangle calculations.

Angle Sum in a Triangle

The Angle Sum in a triangle always amounts to 180 degrees. This basic geometric principle helps determine a missing angle if you know the other two angles. For instance:

For the given triangle, where \( A=110^\text{°} \) and \( C=30^\text{°} \), the missing angle \( B \) is calculated by: \ B = 180^\text{°} - A - C = 180^\text{°} - 110^\text{°} - 30^\text{°} = 40^\text{°} \

• This way, even if only two angles are known, the third angle is easily found.

This fundamental understanding leads to further applications, such as using the Law of Sines or other trigonometric principles to solve for sides or other angles. Make sure always to check your angle sum as a good practice to avoid calculation errors. For non-right triangles, especially, verifying the sum will ensure your other methods are on track.

Triangle Side Calculation

Triangle side calculation can be efficiently managed by combining fundamental principles such as the Law of Sines and the Angle Sum property. The basic steps are:

• Determine the unknown angle.
• Apply the Law of Sines as needed.

For our example:

• First, found \( B=40^\text{°} \) using \( 180^\text{°} - A - C \).

Then, using the Law of Sines, to find side \( a \), we do: \ a = 3 \frac{\text{sin}(110^\text{°})}{\text{sin}(30^\text{°})} ≈ 5.64 \

• Next, to find side \( b \):
\ b = 3 \frac{\text{sin}(40^\text{°})}{\text{sin}(30^\text{°})} ≈ 3.86 \
• This step-by-step assignments of calculating one side and then another simplifies what may initially seem like complex tasks.

Solving triangles by using sound formulae such as the Law of Sines, combined with geometric properties, is better for a logical and straightforward learning process.