Flashcards on Fundamental Mathematics Concepts
Mastering Fundamental Mathematics Concepts: CSCA Guide
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Trigonometry
58 cards
Card 1
Question: How is the graph of y = A sin(ωx + φ) obtained from y = A sin x when φ > 0 or φ < 0?
Answer: Shift the graph of y = A sin x left by φ units if φ > 0, or right by |φ| units if φ < 0, producing y = A sin(x + φ).
Card 2
Question: What effect does replacing x with ωx have on y = sin(x + φ) when forming y = sin(ωx + φ)?
Answer: Scale the x-coordinates by factor 1/ω: if ω > 1 the graph is horizontally compressed (shortened), if 0 < ω < 1 it is horizontally stretched (lengthene
Card 3
Question: What does the amplitude A do to the graph of y = sin(ωx + φ)?
Answer: It scales the y-coordinates by factor A: A > 1 stretches vertically (extends ordinates), 0 < A < 1 compresses them (shortens ordinates).
Card 4
Question: State the sine rule for a triangle relating sides a,b,c and opposite angles A,B,C and the circumradius R.
Answer: a/sin A = b/sin B = c/sin C = 2R.
Card 5
Question: Write the law of cosines for side a in a triangle with sides a,b,c and opposite angle A.
Answer: a^2 = b^2 + c^2 - 2bc cos A (equivalently cos A = (b^2 + c^2 - a^2)/(2bc)).
Card 6
Question: Give the three cosine-law formulas expressing cos A, cos B, and cos C in terms of side lengths.
Answer: cos A = (b^2 + c^2 - a^2)/(2bc); cos B = (a^2 + c^2 - b^2)/(2ac); cos C = (a^2 + b^2 - c^2)/(2ab).
Card 7
Question: Provide the formulas for the area S of a triangle using two sides and the sine of the included angle.
Answer: S = (1/2) a b sin C = (1/2) b c sin A = (1/2) a c sin B.
Card 8
Question: State the cosine addition and subtraction formulas for angles α and β.
Answer: cos(α + β) = cos α cos β − sin α sin β; cos(α − β) = cos α cos β + sin α sin β.
Card 9
Question: What are the sine addition and subtraction formulas for angles α and β?
Answer: sin(α + β) = sin α cos β + cos α sin β; sin(α − β) = sin α cos β − cos α sin β.
Card 10
Question: Write the tangent addition formula in terms of sines and cosines and the simplified form using tangents.
Answer: tan(α + β) = (sin α cos β + cos α sin β)/(cos α cos β − sin α sin β). If cos α cos β ≠ 0: tan(α + β) = (tan α + tan β)/(1 − tan α tan β).