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Questions and Answers
What is the value of sin(2A) if sin(A) = 1/2 and cos(A) = √3/2?
What is the value of sin(2A) if sin(A) = 1/2 and cos(A) = √3/2?
- 1/4 (correct)
- 1/2
- √3/2
- 1/√2
In a right triangle, if the length of the hypotenuse is 5 and the length of one leg is 3, what is the length of the other leg?
In a right triangle, if the length of the hypotenuse is 5 and the length of one leg is 3, what is the length of the other leg?
- 4 (correct)
- 2√2
- 5
- √13
If cos(A) = 2/3, what is the value of cos(A/2)?
If cos(A) = 2/3, what is the value of cos(A/2)?
- ±√((4/9)/2)
- ±√((1/9)/2)
- ±√((5/9)/2) (correct)
- ±√((7/9)/2)
What is the value of cos(2A) if sin(A) = 3/5 and cos(A) = 4/5?
What is the value of cos(2A) if sin(A) = 3/5 and cos(A) = 4/5?
In an oblique triangle, if a = 5, b = 7, and C = 60°, what is the length of c?
In an oblique triangle, if a = 5, b = 7, and C = 60°, what is the length of c?
What is the value of sin(A + B) if sin(A) = 1/2, cos(A) = √3/2, sin(B) = 1/3, and cos(B) = 2/3?
What is the value of sin(A + B) if sin(A) = 1/2, cos(A) = √3/2, sin(B) = 1/3, and cos(B) = 2/3?
What is the value of tan(A) if sin(A) = 2/3 and cos(A) = √5/3?
What is the value of tan(A) if sin(A) = 2/3 and cos(A) = √5/3?
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Study Notes
Trigonometric Identities
- Pythagorean Identity: sin^2(A) + cos^2(A) = 1
- Sum and Difference Identities:
- sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
- sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
- cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
- cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
- Double Angle Identities:
- sin(2A) = 2sin(A)cos(A)
- cos(2A) = cos^2(A) - sin^2(A)
- Half Angle Identities:
- sin(A/2) = ±√((1 - cos(A))/2)
- cos(A/2) = ±√((1 + cos(A))/2)
Solving Triangles
- Right Triangles:
- SOH-CAH-TOA:
- sine = opposite side / hypotenuse
- cosine = adjacent side / hypotenuse
- tangent = opposite side / adjacent side
- Pythagorean Theorem: a^2 + b^2 = c^2 (where c is the hypotenuse)
- SOH-CAH-TOA:
- Oblique Triangles:
- Law of Sines: a / sin(A) = b / sin(B) = c / sin(C)
- Law of Cosines: c^2 = a^2 + b^2 - 2ab * cos(C)
- Heron's Formula: Area = √(s(s-a)(s-b)(s-c)) (where s is the semi-perimeter)
- Solving Triangles Using Identities:
- Use trigonometric identities to rewrite expressions in terms of a single angle or side
- Simplify and solve for the desired angle or side
Trigonometric Identities
- Pythagorean Identity: relates the squares of sine and cosine of an angle to 1
- Sum and Difference Identities: provide formulas for sine and cosine of sums and differences of angles
- Double Angle Identities: relate the sine and cosine of double angles to products of sine and cosine of the original angle
- Half Angle Identities: relate the sine and cosine of half angles to square roots of products of sine and cosine of the original angle
Solving Triangles
Right Triangles
- SOH-CAH-TOA: a mnemonic for remembering the definitions of sine, cosine, and tangent in right triangles
- Pythagorean Theorem: a formula for finding the length of the hypotenuse of a right triangle
Oblique Triangles
- Law of Sines: a formula for finding the lengths of sides or angles of an oblique triangle
- Law of Cosines: a formula for finding the length of a side of an oblique triangle
- Heron's Formula: a formula for finding the area of an oblique triangle
Solving Triangles Using Identities
- Rewriting Expressions: use trigonometric identities to rewrite expressions in terms of a single angle or side
- Simplifying and Solving: simplify the rewritten expressions and solve for the desired angle or side
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