⏳• Module 8 : Advanced Algebra & Trig Mastery Flashcards
Mastering algebraic fractions, quadratics and equations using formulas, all the way to applying trigonometry in 3D.
What is the method for converting between currencies?
Multiply the exchange rate to convert it, and divide to reverse it.
Example : To convert 100 USD to EUR,
1 USD= 0.85 EUR:
100x 0.85 =85 EUR.
How do you prove the quadratic formula by completing the square?
To prove the quadratic formula, start with the general quadratic equation ;
a𝑥²+b𝑥+c=0.
Divide the equation by a, complete the square on x²+(b/a)𝑥, and solve for 𝑥.
How do you factorise quadratics where a ≠ 1?
Use the method of splitting the middle term.
Example : 6x²+5x-6
Split 5x as 6x-x, then factorise:
(2x-3)(3x+2)
How do you multiply algebraic fractions?
Multiply the numerators and denominators separately, then simplify.
Example : 2𝑥/3 x 4/5𝑥 = 8𝑥/15𝑥 = 8/15.
How do you divide algebraic fractions?
Multiply by the reciprocal of the second fraction.
Example : 3𝑥/4 ÷ 5/6𝑥 = 3𝑥/4 x 6𝑥/5 = 18𝑥²/20𝑥 = 9𝑥/10
How do you add algebraic fractions?
Find a common denominator, then add the numerators.
Example : 1/𝑥 + 2/(𝑥+1), common denominator = 𝑥(𝑥+1)
So: (𝑥+1+2𝑥)/𝑥(𝑥+1) = (3𝑥+1)/𝑥(𝑥+1).
How do you subtract algebraic fractions?
Find a common denominator, then subtract the numerators.
Example : 3/(𝑥+2) - 5/𝑥, common denominator = 𝑥(𝑥+2),
So: 3𝑥-5(𝑥+2)/𝑥(𝑥+2) = (-2𝑥-10)/𝑥(𝑥+2).
What is the sine rule?
a/sin A = b/sin B = c/sin C, used in any triangle.
Example : In triangle ABC, if A = 30°, B = 45°, a = 10°, find b.
10/sin 30°=b/sin 45°, solve for b= 10x((sin 45°)/sin 30°).
What is the cosine rule?
c²=a²+b²-2ab x cos C, used in any triangle.
Example : In triangle ABC, a = 7, b = 9, c = 60°, find c.
c²=7²+9² - 2x7x9 x cos 60°, solve for c.
How do you calculate the area of a triangle using sine?
Area = 1/2ab sin C, where an and b are two sides, and C is the included angle.
Example : For a = 6, b = 8, C = 45°,
Area = 1/2 x 6 x 8 x sin 45°
What are the general shapes of sine, cosine and tangent graphs?
Sine: A wave that starts at 0, oscillating between -1 and 1, with a period of 360° (or 2π radians).
Cosine: A wave similar to sine, but starts at 1 and oscillates between -1 and 1.
Tangent: A curve that oscillates between positive and negative infinity, with vertical asymptotes at 90° and 270°.
How do you find the amplitude and period of a sine graph?
Amplitude : The vertical distance from the centerline to the peak.
Period : The horizontal distance for one full cycle of the wave, usually 360° for standard sine functions.
What is the graph of y=2sin(x)? Imagine it.
y=2sin(x)
— We know that the amplitude is 2 (since the coefficient is 2)
— We know that the period is 360° (no change to period).
— So the graph will be the same shape as the sine wave but with a higher amplitude (stretches vertically).
How do you use trigonometry to solve 3D problems?
1 — Use trigonometric ratios (sin, cos, tan) to find missing angles and lengths in 3D shapes.
2 — Apply the Pythagorean theorem to solve for missing sides in right-angled triangles.
How do you know whether to use cosine or sine rules with triangles?
Sine rule : Used when you have either:
— Two angles and one side (AAS or ASA).
— Two sides and a non-included angle (SSA).
Cosine Rule : Used when you have:
— Two sides and the included angle (SAS).
— All three sides of the triangle (SSS).
(The sine rule relates the sides and angles, while the cosine rule connects the sides when you know an angle, or helps find angles angle when all sides are known).
