Question

Given: with altitude h. Two right triangles are formed: one with side lengths , h, and b, and one with side lengths r, h, and a. A diagram of the triangle ABC. A dashed perpendicular line is drawn outside the triangle from the top vertex C to the base. This line joins B. The length of AB is c, AC is b, BC is a the height of the perpendicular line is h and the base is r. Carson starts the proof of the law of cosines with by the definition of the sine ratio and by the definition of the cosine ratio. What are the next steps in the proof? Use the to rewrite each trigonometric equation in terms of the numerator. Then, Carson can write an expression for side in terms of . Next, he can use the to relate a, b, c, and A.

Answer 1

Explanation:

To prove the law of cosines, Carson can proceed with the following steps:

1. Start with the definition of the sine ratio: In a right triangle, the sine of an angle is equal to the length of the side opposite the angle divided by the hypotenuse.

2. Use the definition of the cosine ratio: In a right triangle, the cosine of an angle is equal to the length of the adjacent side divided by the hypotenuse.

3. Rewrite the sine ratio equation in terms of the numerator: Using the given triangle ABC, the sine of angle A can be written as h/b, and the sine of angle B can be written as h/a.

4. Rewrite the cosine ratio equation in terms of the numerator: The cosine of angle A can be written as b/r, and the cosine of angle B can be written as a/r.

5. Write an expression for side c in terms of h, a, and b: Using the Pythagorean theorem in triangle ABC, we can write c^2 = a^2 + b^2.

6. Use the Pythagorean identity: The Pythagorean identity states that sin^2θ + cos^2θ = 1. By applying this identity to the equations obtained in steps 3 and 4, Carson can rewrite them in terms of h, a, and b.

7. Substitute the rewritten equations from step 6 into the expression for side c from step 5. This will give an expression for side c in terms of h, a, and b.

8. Use the Law of Sines: The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. Carson can use this law to relate side a, side b, side c, and angle A.

By following these steps, Carson can establish the law of cosines and further explore the relationship between the sides and angles in triangle ABC.