Podcast
Questions and Answers
Which of the following is a Pythagorean identity?
Which of the following is a Pythagorean identity?
The area of a triangle can be calculated using the formula Area = (1/2)ab * cos(C).
The area of a triangle can be calculated using the formula Area = (1/2)ab * cos(C).
False
What is the double angle formula for sine?
What is the double angle formula for sine?
sin(2x) = 2sin(x)cos(x)
The law of sines states that a/sin(A) = b/sin(B) = c/sin(C). Fill in the blank. The ratio of side 'b' over sin(B) is equal to _____ over sin(A).
The law of sines states that a/sin(A) = b/sin(B) = c/sin(C). Fill in the blank. The ratio of side 'b' over sin(B) is equal to _____ over sin(A).
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Match the trigonometric identities with their corresponding formulas:
Match the trigonometric identities with their corresponding formulas:
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Study Notes
Trigonometric Identities
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Reciprocal Identities: These relate trigonometric functions to each other.
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sin(x) = 1/csc(x), cos(x) = 1/sec(x), tan(x) = 1/cot(x)
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csc(x) = 1/sin(x), sec(x) = 1/cos(x), cot(x) = 1/tan(x)
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Pythagorean Identities: Based on the Pythagorean theorem, these connect sine and cosine.
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sin²(x) + cos²(x) = 1
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1 + tan²(x) = sec²(x)
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1 + cot²(x) = csc²(x)
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Quotient Identities: These define tangent and cotangent in terms of sine and cosine.
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tan(x) = sin(x)/cos(x)
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cot(x) = cos(x)/sin(x)
Law of Cosines & Law of Sines
- Law of Cosines: Used to find sides and angles in triangles.
- c² = a² + b² - 2ab * cos(C)
- Law of Sines: Another useful tool for finding unknown sides and angles.
- a/sin(A) = b/sin(B) = c/sin(C)
Area of a Triangle (Trigonometric Form)
- Calculating the area of a triangle using trigonometric functions.
- Area = (1/2)ab * sin(C)
Sum and Difference Formulas
- Sine:
- sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
- Cosine:
- cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B)
- Tangent:
- tan(A ± B) = [tan(A) ± tan(B)] / [1 ∓ tan(A)tan(B)]
Double Angle Formulas
- Sine:
- sin(2x) = 2sin(x)cos(x)
- Cosine:
- cos(2x) = cos²(x) - sin²(x) = 2cos²(x) - 1 = 1 - 2sin²(x)
- Tangent:
- tan(2x) = 2tan(x) / [1 - tan²(x)]
Half-Angle Formulas
- Sine:
- sin(x/2) = ±√[(1 - cos(x))/2]
- Cosine:
- cos(x/2) = ±√[(1 + cos(x))/2]
- Tangent:
- tan(x/2) = ±√[(1 - cos(x))/(1 + cos(x))]
- tan(x/2) = sin(x) / [1 + cos(x)] or [1 - cos(x)] / sin(x)
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Description
This quiz covers essential trigonometric identities including reciprocal, Pythagorean, and quotient identities. Additionally, it explores the Law of Cosines and Law of Sines, as well as calculating the area of triangles using trigonometric functions. Test your understanding of these fundamental concepts in trigonometry!
