Problem 19 - Algebra and Trigonometry

Question

Solve each triangle. $$ A=55^{\circ}, \quad B=25^{\circ}, \quad a=4 $$

Step-by-Step Solution

Answer

C = 100^\text{°}, b \approx 2.06, c \approx 4.80

Step 1: Find Angle C

The sum of the interior angles in a triangle is always 180 degrees. Use this information to find angle C: \[ C = 180^\text{°} - A - B \] Substitute the given values: \[ C = 180^\text{°} - 55^\text{°} - 25^\text{°} = 100^\text{°} \]

Step 2: Use the Law of Sines to Find Side b

The Law of Sines states: \[ \frac{a}{\sin A} = \frac{b}{\sin B} \] Rearrange to solve for b: \[ b = a \cdot \frac{\sin B}{\sin A} \] Substitute the known values: \[ b = 4 \cdot \frac{\sin 25^\text{°}}{\sin 55^\text{°}} \] Use a calculator to find the sines and compute b: \[ b \approx 4 \cdot \frac{0.4226}{0.8192} \approx 2.06 \]

Step 3: Use the Law of Sines to Find Side c

Again, apply the Law of Sines: \[ \frac{a}{\sin A} = \frac{c}{\sin C} \] Rearrange to solve for c: \[ c = a \cdot \frac{\sin C}{\sin A} \] Substitute the known values: \[ c = 4 \cdot \frac{\sin 100^\text{°}}{\sin 55^\text{°}} \] Use a calculator to find the sines and compute c: \[ c \approx 4 \cdot \frac{0.9848}{0.8192} \approx 4.80 \]

Key Concepts

Interior Angles

In any triangle, the sum of the interior angles is always equal to 180 degrees. This means that if you know two angles, you can easily find the third. For example, if you have angles A and B, you can find angle C using the formula: \[ C = 180^\text{°} - A - B \] . In the given exercise, we have: \[ A = 55^\text{°}, \ B = 25^\text{°}, \ C = 100^\text{°}\] . It's always useful to remember this fundamental property, as it helps simplify various geometric problems. Breaking down complex problems into simpler parts often makes them more solvable.

Law of Sines

The Law of Sines is essential for solving triangles, especially non-right-angled ones. It states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides. Mathematically, this is given by: \[ \frac{a}{\text{sin} \ A} = \frac{b}{\text{sin} \ B} = \frac{c}{\text{sin} \ C} \] . In our exercise, we use the Law of Sines to find sides 'b' and 'c'. For side 'b', the equation becomes: \[ \frac{a}{\text{sin} \ A} = \frac{b}{\text{sin} \ B} \] . Rearranging, we get: \[ b = a \ \frac{\text{sin} \ B}{\text{sin} \ A} \] . Substituting the values, we find: \[ b = 4 \ \frac{\text{sin} \ 25^\text{°}}{\text{sin} \ 55^\text{°}} \ \text{which calculates to} \ \ b ≈ 2.06 \] . Similarly, for side 'c', we use: \[ c = a \frac{\text{sin} \ C}{\text{sin} \ A} \] , resulting in: \[ c = 4 \ \frac{\text{sin} \ 100^\text{°}}{\text{sin} \ 55^\text{°}} \ \text{which calculates to} \ \ c ≈ 4.80 \] .

Trigonometric Functions

Trigonometric functions such as sine (sin), cosine (cos), and tangent (tan) are fundamental in triangle solving. In this exercise, we specifically use the sine function. The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse: \[ \text{sin}( \ \theta ) = \frac{\text{opposite}}{\text{hypotenuse}} \] . For example, \[ \text{sin} \ 55^\text{°} ≈ 0.8192 \] and \[ \text{sin} \ 25^\text{°} ≈ 0.4226 \] . These values are essential when applying the Law of Sines. Knowing how to find and use these trigonometric functions is crucial for solving triangles effectively.

Angle Sum Property

The angle sum property is a straightforward but powerful rule in geometry. It states that the sum of the internal angles of a triangle is always 180 degrees. If you are given two angles, you can always find the third one. For instance, in the given exercise with: \[ A = 55^\text{°}, \ B = 25^\text{°} \] , we determine angle C by: \[ C = 180^\text{°} - A - B \] which gives: \[ C = 100^\text{°} \] . This property helps in verifying the correctness of your solutions and simplifying the process of solving triangles. It is essential to always keep this property in mind and apply it where necessary.