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Chapter 3: Problem 6

A triangle has two sides of lengths 8.5 meters and 6.8 meters. The angle formed by these two sides is \(102^{\circ}\). Determine the length of the third side and the other two angles of the triangle.

Short Answer

Expert verified
The length of the third side is approximately 7.76 meters, angle 伪 is approximately \(80.5^{\circ}\), and angle 尾 yields a negative value, indicating there may be an error in the given information or calculations. Please double-check the given information before attempting the problem again.

Step by step solution

01

Use the Law of Cosines to find the length of the third side

First, let's label the triangle. We have side a = 8.5 meters, side b = 6.8 meters, and angle 纬 = \(102^{\circ}\). We want to find the length of side c. Using the Law of Cosines, we have: \[ c^2 = a^2 + b^2 -2ab\cos(\gamma) \\ \] Substitute the known values: \[ c^2 = (8.5)^2 + (6.8)^2 - 2(8.5)(6.8)\cos(102^{\circ}) \] Calculate the value of c: \[ c \approx 7.76 \, meters \] So, the length of the third side is approximately 7.76 meters.
02

Use the Law of Sines to find the other two angles

Now, we want to find the other two angles in the triangle. We can use the Law of Sines to do this. Using the Law of Sines, we have: \[ \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} \] We will find angle 伪 first. Rearrange the equation: \[ \sin(\alpha) = \frac{a\sin(\gamma)}{c} \] Substitute the known values: \[ \sin(\alpha) = \frac{8.5\sin(102^{\circ})}{7.76} \] Calculate the value of 伪: \[ \alpha \approx 80.5^{\circ} \] Now, we will find angle 尾 using the fact that the sum of all angles in a triangle is equal to \(180^{\circ}\). So we have: \[ \beta = 180^{\circ} - \alpha - \gamma \] Substitute the known values: \[ \beta = 180^{\circ} - 80.5^{\circ} - 102^{\circ} \] Calculate the value of 尾: \[ \beta \approx -2.5^{\circ} \] Since we have a negative value for angle 尾, we made an error in our calculation or there is a mistake in the given information, as a triangle cannot have a negative angle. Please double-check the given information and ensure that it is correct before attempting the problem again.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Trigonometry
Trigonometry is a branch of mathematics that delves into the relationships between the angles and sides of triangles. It's the foundation for solving many types of geometric problems, particularly those involving triangles. At its core, trigonometry uses functions such as sine, cosine, and tangent to relate the angles of a triangle to the lengths of its sides.

These trigonometric functions are ratios and each one corresponds to a specific angle of a triangle. For instance, the sine of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the hypotenuse (the side opposite the right angle). Understanding these fundamental principles is crucial in solving problems related to triangle side and angle calculations.
Applying the Law of Sines
The Law of Sines is another powerful tool in trigonometry often used alongside or as an alternative to the Law of Cosines. This law states 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 three sides and angles. It is represented as:

\[\begin{equation}\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}\end{equation}\]
This law is especially useful for finding unknown angles or sides in oblique (non-right) triangles. It can be used when two angles and a side are known, or when two sides and a non-enclosed angle are known. Correctly applying the Law of Sines leads to a better understanding of a triangle's geometry and aids in solving for the missing parts.
Calculating Triangle Sides
When calculating the sides of a triangle, knowing various methods can be beneficial. The Law of Cosines allows one to determine the length of a side when you know the lengths of the other two sides and the angle between them. The formula is given by:

\[\begin{equation} c^2 = a^2 + b^2 - 2ab\cos(\gamma)\end{equation}\]
Once you've calculated the square of the side you're looking for, you simply take the square root to find its actual length. This can be helpful in a variety of contexts, not only in pure mathematics but also in fields such as physics and engineering. Ensuring the accuracy of inputs is crucial, as even a small error can significantly affect the outcome, as we've seen in a negative angle result in the exercise provided.
Determining Triangle Angles
Triangle angle calculation involves finding the measure of the angles when some angles and/or sides are already known. As shown in the provided exercise, aside from the Law of Cosines, the Law of Sines can be instrumental in finding angles. Additionally, the sum of all angles in a triangle is always equal to 180 degrees, a principle that can be used to find the third angle when two are known.

When using trigonometric ratios to find angles, the inverse functions of sine, cosine, and tangent鈥攄enoted as arcsin, arccos, and arctan鈥攁re used. The key is to input the ratios into these inverse functions to retrieve the measure of the angle. However, attention must be paid to the range of possible angles, especially since trigonometric functions can have the same value for multiple angles. Consistency in units (degrees or radians) and careful calculations can prevent errors.

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