This video tutorial covers the derivation of the laws of sines and cosines. It begins with an introduction to the topic, followed by a detailed step-by-step derivation of the law of sines using right triangles and trigonometric identities. The video then transitions to deriving the law of cosines, employing the Pythagorean theorem and algebraic manipulation. The tutorial concludes with a summary and a call to action for viewers to engage with the content by liking, subscribing, and sharing.

What is the purpose of labeling the altitude in the triangle?

• To complicate the geometric structure

• To increase the number of sides in the triangle

• To change the angles of the triangle

• To create right triangles for applying trigonometric functions

Which trigonometric function is used to relate the opposite side and the hypotenuse in a right triangle?

• Cosine

• Tangent

• Cotangent

• Sine

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CCSS.HSG.SRT.C.6

What does the equation 'sin of a = H2 over C' imply about angle a?

• It is not relevant to the triangle's properties

• It is the smallest angle in the triangle

• It is opposite to side C

• It is calculated using the altitude H2

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CCSS.HSG.SRT.C.7

How is the cosine of an angle defined in the context of these triangles?

• Adjacent over hypotenuse

• Opposite over hypotenuse

• Adjacent over opposite

• Hypotenuse over adjacent

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CCSS.HSG.SRT.C.6

Which law is derived by setting equations for H2 equal and eliminating H2?

• Law of cosines

• Law of cotangents

• Law of sines

• Law of tangents

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What is the result of substituting 'c^2' into the second equation during the cosine law derivation?

• Simplifies the equation to find x

• Eliminates the variable H1

• Creates a new triangle

• None of the above

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CCSS.HSG.SRT.D.10

CCSS.HSG.SRT.D.11

What does the cosine law 'b^2 = a^2 + c^2 - 2ac cos(B)' describe?

• The sum of all angles in the triangle

• The calculation of the hypotenuse

• The relationship between all sides and angle B

• None of the above

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CCSS.HSG.SRT.D.10

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What is the significance of the altitude in deriving the cosine laws?

• It determines the type of triangle

• It is used to calculate the area of the triangle

• It helps in dividing the sides into measurable segments

• It changes the triangle's angles

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CCSS.HSG.SRT.D.10

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How are the laws of sines and cosines useful in geometry?

• They help in calculating angles and sides in non-right triangles

• They are only applicable to right triangles

• They determine the radius of circumscribed circles

• They are used to calculate the perimeter of triangles

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CCSS.HSG.SRT.D.10

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What is the final cosine law derived in the video?

• b^2 = a^2 + c^2 - 2ac cos(B)

• c^2 = a^2 + b^2 - 2ab cos(C)

• a^2 = b^2 + c^2 - 2bc cos(A)

• None of the above

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CCSS.HSG.SRT.D.10

CCSS.HSG.SRT.D.11