Problem 9 - Precalculus

Question

Solve the following triangles with the given measures. $$\alpha=46^{\circ}, \gamma=72^{\circ}, b=200 \mathrm{cm}$$

Step-by-Step Solution

Answer

\( a \approx 162.9 \text{ cm} \), \( b = 200 \text{ cm} \), \( c \approx 215.5 \text{ cm} \).

Step 1: Find the third angle

In any triangle, the sum of the interior angles is always 180 degrees. We can use this property to find angle \( \beta \) since the other two angles are provided. Use the formula:\[ \beta = 180^{\circ} - \alpha - \gamma \]Substituting the given angles:\[ \beta = 180^{\circ} - 46^{\circ} - 72^{\circ} = 62^{\circ} \]Thus, \( \beta = 62^{\circ} \).

Step 2: Solve for side a using the Law of Sines

The Law of Sines states that \( \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} \). Since we know \( b \), \( \alpha \), \( \beta \), and want to find \( a \), rearrange the formula to solve for \( a \):\[ a = \frac{b \cdot \sin(\alpha)}{\sin(\beta)} \]Substitute the given values:\[ a = \frac{200 \cdot \sin(46^{\circ})}{\sin(62^{\circ})} \]Calculate \( a \).

Step 3: Solve for side c using the Law of Sines

With the Law of Sines and having \( \gamma \), solve for \( c \):\[ c = \frac{b \cdot \sin(\gamma)}{\sin(\beta)} \]Plug in the known values:\[ c = \frac{200 \cdot \sin(72^{\circ})}{\sin(62^{\circ})} \]Calculate \( c \).

Step 4: Calculate the final answers

Let's compute the values reached in Steps 2 and 3 using a calculator with trigonometric functions. For side \( a \):\[ a \approx \frac{200 \cdot 0.7193}{0.8829} = 162.9 \text{ cm} \]For side \( c \):\[ c \approx \frac{200 \cdot 0.9511}{0.8829} = 215.5 \text{ cm} \]Hence, the sides of the triangle are approximately \( a = 162.9 \text{ cm} \), \( b = 200 \text{ cm} \), and \( c = 215.5 \text{ cm} \).

Key Concepts

Law of Sines

The **Law of Sines** is a fundamental principle in trigonometry used to relate the angles and sides of a triangle. It is especially useful for solving oblique triangles, which are triangles without a right angle. The Law is expressed as:\[\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}\]This formula suggests that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and angles in the triangle.
To apply the law:

• First identify the known angle-side pairs. In this problem, side \( b \) and angle \( \beta \) are used.
• Then choose the unknown side and corresponding angle for which you need to find the length.
• Rearrange the formula to solve for the desired side.
• Substitute the known values and calculate using trigonometric functions.

The Law of Sines allows for efficient calculation of unknowns when dealing with non-right triangles.

angles in a triangle

The sum of all angles in a triangle always adds up to 180 degrees. This is a basic property of triangles and is crucial when trying to find a missing angle.
In the given problem, two angles \( \alpha = 46^{\circ} \) and \( \gamma = 72^{\circ} \) are known. Using the formula:\[\beta = 180^\circ - \alpha - \gamma\]Substitute the known angles into the formula:\[\beta = 180^\circ - 46^\circ - 72^\circ = 62^\circ\]This calculation identifies \( \beta \), the third angle, as \( 62^\circ \).
This property simplifies calculating triangles with missing angles and is a key step when using trigonometric laws such as the Law of Sines or the Law of Cosines. Always remember: knowing just two angles in a triangle allows for the determination of the third.

solving triangles

Solving a triangle means finding the lengths of its sides and the measures of its angles. This exercise involved solving a triangle using given angles and one side.
The process begins by using the angle sum property to find any missing angles. Once all angles are known:

• The Law of Sines becomes a powerful tool for finding unknown side lengths. Use it to form equations that relate known and unknown quantities.
• Plug known values into the Law of Sines and solve for the unknown sides. This often involves using a calculator to evaluate sine values.
• Compute side lengths using rearranged versions of the Law of Sines formula.

Finally, verify your results by checking that calculated side lengths make sense in the context of the triangle. The technique and laws used here ensure the triangle's dimensions are consistent, thus completing the triangle solving process.