Problem 100 - Algebra and Trigonometry

Question

Solve the triangle: \(A=65^{\circ}, B=37^{\circ}, c=10\).

Step-by-Step Solution

Answer

The angles are \(A = 65^{\text{o}}, B = 37^{\text{o}}, C = 78^{\text{o}}\), with sides \(a \approx 9.27\) and \(b \approx 6.15\).

Step 1: Calculate angle C

Use the fact that the sum of the angles in any triangle is always \(180^{\text{o}}\). Given \(A = 65^{\text{o}}\) and \(B = 37^{\text{o}}\), calculate angle C using the equation:\[ C = 180^{\text{o}} - A - B \]Substitute the known values:\[ C = 180^{\text{o}} - 65^{\text{o}} - 37^{\text{o}} = 78^{\text{o}} \]

Step 2: Use the Law of Sines

Apply the Law of Sines to find the lengths of sides a and b. The Law of Sines is given by:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]Given \(c = 10\), \(A = 65^{\text{o}}\), \(B = 37^{\text{o}}\), and \(C = 78^{\text{o}}\), set up the ratios:\[ \frac{a}{\sin 65^{\text{o}}} = \frac{10}{\sin 78^{\text{o}}} \]\[ \frac{b}{\sin 37^{\text{o}}} = \frac{10}{\sin 78^{\text{o}}} \]

Step 3: Solve for side a

Rearrange the equation to solve for side a:\[ a = 10 \cdot \frac{\sin 65^{\text{o}}}{\sin 78^{\text{o}}} \]Calculate using approximate values:\[ a \approx 10 \cdot \frac{0.9063}{0.9781} \approx 9.27 \]

Step 4: Solve for side b

Similarly, rearrange the equation to solve for side b:\[ b = 10 \cdot \frac{\sin 37^{\text{o}}}{\sin 78^{\text{o}}} \]Calculate using approximate values:\[ b \approx 10 \cdot \frac{0.6018}{0.9781} \approx 6.15 \]

Key Concepts

Triangle Angles

Understanding the angles within a triangle is crucial to solving it. A triangle is a three-sided polygon, and the sum of its interior angles is always 180 degrees. This is a fundamental property known as the Triangle Angle Sum Theorem. For example, if you know two angles of a triangle, you can easily find the third angle by subtracting the sum of the known angles from 180 degrees. In our given triangle exercise, we know angles A and B to be 65 degrees and 37 degrees, respectively. By using the equation:\[ C = 180^{\text{o}} - A - B \]we substitute the given values:\[ C = 180^{\text{o}} - 65^{\text{o}} - 37^{\text{o}} = 78^{\text{o}} \]Thus, angle C is 78 degrees.

Law of Sines

The Law of Sines is a powerful trigonometric tool used for solving triangles, particularly when you know one angle and its opposite side. It relates the lengths of the sides of a triangle to the sines of its angles. The Law of Sines formula is:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]In our exercise, we have a non-right angle triangle with angles and sides partially known: angles A = 65 degrees, B = 37 degrees, C = 78 degrees, and side c = 10 units. By setting up the ratios:\[ \frac{a}{\sin 65^{\text{o}}} = \frac{10}{\sin 78^{\text{o}}} \]\[ \frac{b}{\sin 37^{\text{o}}} = \frac{10}{\sin 78^{\text{o}}} \]The key is to cross-multiply and solve these ratios to find the missing sides a and b.

Trigonometry Calculations

Trigonometry deals with the relationships between angles and sides in triangles. Calculations often involve using known values and trigonometric functions like sine, cosine, and tangent. In our exercise, to find the side lengths a and b, we use the approximations of sines for our known angles. For angle A (65 degrees), sine is approximately 0.9063, and for angle C (78 degrees), it is approximately 0.9781. To solve for side a, we rearrange and calculate as follows:\[ a = 10 \cdot \frac{\sin 65^{\text{o}}}{\sin 78^{\text{o}}} \]\[ a \approx 10 \cdot \frac{0.9063}{0.9781} \approx 9.27 \]Similarly, for angle B (37 degrees), sine is approximately 0.6018. To solve for side b, we rearrange and calculate:\[ b = 10 \cdot \frac{\sin 37^{\text{o}}}{\sin 78^{\text{o}}} \]\[ b \approx 10 \cdot \frac{0.6018}{0.9781} \approx 6.15 \]These calculated side lengths help us to solve the triangle completely.

Triangle Sides

Determining the lengths of the sides of a triangle involves knowing the relationship between the angles and the sides. In our given exercise, we start with one known side 'c' and two known angles A and B. By using the Law of Sines and trigonometric calculations, we find the other two sides a and b. The sides of a triangle are directly related to the sines of their opposite angles. This relationship is harnessed to find unknown side lengths accurately. Thus, solving for side a:\[ a = 10 \cdot \frac{\sin 65^{\text{o}}}{\sin 78^{\text{o}}} \approx 9.27 \]and solving for side b:\[ b = 10 \cdot \frac{\sin 37^{\text{o}}}{\sin 78^{\text{o}}} \approx 6.15 \]The goal is to ensure all angles and sides are accounted for, giving a complete picture of the triangle's dimensions.