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Flashcards: Prove Laws of Sines and Cosines: CCSS.Math.Content.HSG-SRT.D.10

The Law of Cosines is a generalization of which common theorem?

Pythagorean Theorem

Fundamental Theorem of Algebra

Triangle Sum Theorem

Triangle Proportionality Theorem

Side Angle Side Similarity Theorem

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Flashcards: Prove Laws of Sines and Cosines: CCSS.Math.Content.HSG-SRT.D.10

The Law of Cosines is a generalization of which common theorem?

Your answer:

Fundamental Theorem of Algebra

Your answer:

Triangle Sum Theorem

Your answer:

Triangle Proportionality Theorem

Your answer:

Side Angle Side Similarity Theorem

Correct answer:

Pythagorean Theorem

Explanation:

The Law of Cosines is a generalization of the Pythagorean Theorem. To better understand this, start by drawing triangle ABC with sides a, b, and c. Then, construct the altitude from point A to get height h. Finally label the distance between point B and the point where h meets side a as distance r.

Screen shot 2020 07 01 at 7.58.05 pm

We will arbitrarily work to solve for side b, but we could do the same for a and c (after drawing the correct altitudes), which is why we can generalize that the other two formulas to find a and c follow.

By definition, we know that $\sin\angle B= \frac{h}{c}$ . If we multiply both sides by c, we get $c\sin\angle B= h$ .

Also, we know that $\cos\angle B= \frac{r}{c}$ . If we multiply both sides by c, we get $c\cos\angle B= r$ .

Using the Pythagorean Theorem, we get b^2=h^2+(a-r)^2 .

Substituting in h and r from above, we get:

$b^2=(c\sin \angle B)^2+ (a-c\cos\angle B)^2$

$= c^2\sin^2 \angle B + a^2-2ac\cos\angle B + c^2\cos^2\angle B$

$= c^2 (sin^2 \angle B+cos^2\angle B) + a^2-2ac\cos\angle B$ (because $sin^2 \angle B+cos^2\angle B=1$ )

$= c^2 + a^2-2ac\cos\angle B$

Therefore $b^2= c^2 + a^2-2ac\cos\angle B$ .

The other two versions of Law of Cosines are:

$a^2= b^2 + c^2-2bc\cos\angle A$

$c^2= a^2 + b^2-2ab\cos\angle C$

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All Common Core: High School - Geometry Resources

Flashcards: Prove Laws of Sines and Cosines: CCSS.Math.Content.HSG-SRT.D.10

Flashcards: Prove Laws of Sines and Cosines: CCSS.Math.Content.HSG-SRT.D.10

Flashcards: Prove Laws of Sines and Cosines: CCSS.Math.Content.HSG-SRT.D.10

Flashcards: Prove Laws of Sines and Cosines: CCSS.Math.Content.HSG-SRT.D.10

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