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Chapter 9: Problem 69

State the Law of Cosines in words.

Short Answer

Expert verified
The Law of Cosines states: The square of one side of a triangle is equal to the sum of the squares of the other two sides minus two times the product of these sides and the cosine of the included angle.

Step by step solution

01

Identify the Law of Cosines

The Law of Cosines is a formula used to relate the lengths of the sides of a triangle to the cosine of one of its angles.
02

Express the Law of Cosines in Words

The formula states that for any triangle, the square of one side equals the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the included angle.
03

Form the Complete Statement

In any triangle, the square of one side is equal to the sum of the squares of the other two sides, minus two times the product of the other two sides and the cosine of the included angle.

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Key Concepts

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

trigonometry
Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It is essential for solving problems involving triangles, especially when the angles and sides are not perpendicular. Key concepts in trigonometry include sine, cosine, and tangent, which relate angles to side lengths in right-angled triangles. For non-right triangles, the Laws of Sines and Cosines are critical.
triangle sides
In a triangle, each side has a unique relationship with the other sides and angles. Knowing the lengths of two sides and the angle between them can help you find the length of the third side. Using the Law of Cosines, you can determine an unknown side or angle within any triangle, not just right-angled triangles. This law is very useful in various applications, including physics, engineering, and astronomy.
angle cosine relationship
The cosine of an angle in a triangle describes the relationship between the lengths of the sides of the triangle. According to the Law of Cosines, for any triangle with sides a, b, and c and an included angle C: equation: \[c^2 = a^2 + b^2 - 2ab \cos(C)\]This formula shows that the square of side c is the sum of the squares of the other two sides minus twice their product times the cosine of the included angle. This relationship is extremely helpful for finding unknown side lengths or angles in a triangle.
geometric formulas
Geometric formulas are equations that relate various dimensions of geometric shapes. In the context of triangles, the most common formulas are the Law of Sines and the Law of Cosines. These formulas allow you to solve for unknown angles and sides. The Law of Cosines is particularly useful when dealing with non-right triangles. By understanding and applying these formulas, you can find missing side lengths and angles, providing a comprehensive understanding of triangle properties.

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Most popular questions from this chapter

(a) use the Product-to-Sum Formulas to express each product as a sum, and (b) use the method of adding \(y\) -coordinates to graph each function on the interval \([0,2 \pi] .\) $$ G(x)=\cos (4 x) \cos (2 x) $$

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Write the equation \(100=a^{0.2 x}\) in logarithmic form.

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An object of mass \(m\) (in grams) attached to a coiled spring with damping factor \(b\) (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is \(T\) (in seconds) under simple harmonic motion. (a) Find a function that relates the displacement d of the object from its rest position after \(t\) seconds. (b) Graph the function found in part (a) for 5 oscillations using a graphing utility. $$ m=10, \quad a=5, \quad b=0.8, \quad T=3 $$
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