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Common Core: High School - Geometry : Prove Laws of Sines and Cosines: CCSS.Math.Content.HSG-SRT.D.10

Study concepts, example questions & explanations for Common Core: High School - Geometry

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Example Questions

Example Question #168 : High School: Geometry

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

Possible Answers:

Triangle Proportionality Theorem

Pythagorean Theorem

Fundamental Theorem of Algebra

Side Angle Side Similarity Theorem

Triangle Sum 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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Example Question #169 : High School: Geometry

True or false: The Law of Sines applies to all types of triangles.

Possible Answers:

False

True

Correct answer:

True

Explanation:

This is true. While the Pythagorean Theorem only applies to right triangles, the Law of Sines applies to all triangles no matter if they are right, acute, or obtuse.

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Example Question #170 : High School: Geometry

True or False: The Law of Cosines applies to all triangles.

Possible Answers:

True

False

Correct answer:

True

Explanation:

This is true. While the Pythagorean Theorem only applies to right triangles, the Law of Cosines applies to all triangles no matter if they are right, acute, or obtuse.

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Example Question #61 : Similarity, Right Triangles, & Trigonometry

Choose the answer choice that correctly states the Law of Sines, given the triangle below.

Screen shot 2020 08 19 at 4.54.09 pm

Possible Answers:

$A^{2}+B^{2}-2AB\sin (\gamma )=C^{2}$

$\frac{A}{\sin (\gamma )}=\frac{B}{\sin (\alpha )}=\frac{C}{\sin (\beta )}$

$A^{2}+B^{2}-2AB\cos (\gamma )=C^{2}$

$\frac{A}{\cos (\gamma )}=\frac{B}{\cos (\alpha )}=\frac{C}{\cos (\beta )}$

Correct answer:

$\frac{A}{\sin (\gamma )}=\frac{B}{\sin (\alpha )}=\frac{C}{\sin (\beta )}$

Explanation:

The Law of Sines uses ratios of a triangle’s sides and angles to allow us to be able to solve for unknown sides and/or angles. This relationship is true for any triangle.

Screen shot 2020 08 19 at 4.54.09 pm

The Law of Sines is $\frac{A}{\sin (\gamma )}=\frac{B}{\sin (\alpha )}=\frac{C}{\sin (\beta )}$ .

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Example Question #62 : Similarity, Right Triangles, & Trigonometry

Which of the following expressions correctly states the Law of Cosines? Use the following triangle for reference.

Screen shot 2020 08 20 at 10.56.17 am

Possible Answers:

$A^{2}+B^{2}-2AB\cos (\gamma )=C^{2}$

$A^{2}+B^{2}-2AB\sin (\gamma )=C^{2}$

$\frac{A}{\cos (\gamma )}=\frac{B}{\cos (\alpha )}=\frac{C}{\cos (\beta )}$

$\frac{A}{\sin (\gamma )}=\frac{B}{\sin (\alpha )}=\frac{C}{\sin (\beta )}$

Correct answer:

$A^{2}+B^{2}-2AB\cos (\gamma )=C^{2}$

Explanation:

The Law of Cosines should look similar to a theorem we use quite commonly for right triangles. The Law of Cosines is basically the Pythagorean Theorem, but it has an added argument so that it is true for any and all triangles.

Screen shot 2020 08 20 at 10.56.17 am

The Law of Cosines is $A^{2}+B^{2}-2AB\cos (\gamma )=C^{2}$

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Example Question #63 : Similarity, Right Triangles, & Trigonometry

Choose the answer choice that correctly states the Law of Sines, given the triangle below.

Screen shot 2020 08 20 at 11.04.58 am

Possible Answers:

$A^{2}+B^{2}-2AB\cos (\gamma )=C^{2}$

$A^{2}+B^{2}-2AB\sin (\gamma )=C^{2}$

$\frac{A}{\sin (\gamma )}=\frac{B}{\sin (\alpha )}=\frac{C}{\sin (\beta )}$

$\frac{A}{\cos (\gamma )}=\frac{B}{\cos (\alpha )}=\frac{C}{\cos (\beta )}$

Correct answer:

$\frac{A}{\sin (\gamma )}=\frac{B}{\sin (\alpha )}=\frac{C}{\sin (\beta )}$

Explanation:

The Law of Sines uses ratios of a triangle’s sides and angles to allow us to be able to solve for unknown sides and/or angles. This relationship is true for any triangle.

Screen shot 2020 08 20 at 11.04.58 am

The Law of Sines is $\frac{A}{\sin (\gamma )}=\frac{B}{\sin (\alpha )}=\frac{C}{\sin (\beta )}$ .

Report an Error

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Common Core: High School - Geometry : Prove Laws of Sines and Cosines: CCSS.Math.Content.HSG-SRT.D.10

Study concepts, example questions & explanations for Common Core: High School - Geometry

All Common Core: High School - Geometry Resources

Example Questions

Example Question #168 : High School: Geometry

Example Question #169 : High School: Geometry

Example Question #170 : High School: Geometry

Example Question #61 : Similarity, Right Triangles, & Trigonometry

Example Question #62 : Similarity, Right Triangles, & Trigonometry

Example Question #63 : Similarity, Right Triangles, & Trigonometry

All Common Core: High School - Geometry Resources

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