Untitled Flashcards Set

Right-Angled Trigonometry

• Sine, Cosine, and Tangent Ratios:

sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}

cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}

tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

• Pythagorean Theorem (for solving sides):

a2+b2=c2a^2 + b^2 = c^2

Coordinate Geometry

• Distance Formula:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

• Midpoint Formula:

M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

• Gradient (Slope) Formula:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

• Equation of a Straight Line:

y=mx+cy = mx + c

y−y1=m(x−x1)y - y_1 = m(x - x_1)

Sequences & Series

• Arithmetic Sequence:

an=a+(n−1)da_n = a + (n - 1)d

• Sum of Arithmetic Series:

Sn=n2(2a+(n−1)d)S_n = \frac{n}{2} (2a + (n - 1)d)

Sn=n2(a+l)S_n = \frac{n}{2} (a + l)

• Geometric Sequence:

an=a⋅r(n−1)a_n = a \cdot r^{(n - 1)}

• Sum of Geometric Series:

Sn=a(1−rn1−r)(for r≠1)S_n = a \left( \frac{1 - r^n}{1 - r} \right) \quad \text{(for } r \neq 1\text{)}

S∞=a1−r(for ∣r∣<1)S_\infty = \frac{a}{1 - r} \quad \text{(for } |r| < 1\text{)}

Variation & Power Modelling

• Direct Variation:

y=kxy = kx

• Inverse Variation:

y=kxy = \frac{k}{x}

• Power Relationships:

y=kxny = kx^n

Line of Best Fit & Regression

• Correlation Coefficient (r): Measures the strength and direction of a linear relationship between two variables.

• Equation of the Line of Best Fit: Typically found using linear regression methods.

Linear Programming

• Formulating Linear Inequalities: Based on constraints given in word problems.

• Graphing Inequalities: Plotting lines and shading feasible regions.

• Optimization: Finding the maximum or minimum value of an objective function within the feasible region.

Trigonometric Graphs & Modelling

• Sine, Cosine, and Tangent Functions:

y=sin⁡(x),y=cos⁡(x),y=tan⁡(x)y = \sin(x), \quad y = \cos(x), \quad y = \tan(x)

• Amplitude and Period:

  • Amplitude: The maximum value of the sine or cosine function.

  • Period: The length of one complete cycle of the function.

Additional Mathematics (AMath) Topics

Linear Law

• Transforming Non-Linear Equations:

y=axb  ⟹  log⁡(y)=log⁡(a)+blog⁡(x)y = ax^b \implies \log(y) = \log(a) + b\log(x)

y=aebx  ⟹  ln⁡(y)=ln⁡(a)+bxy = ae^{bx} \implies \ln(y) = \ln(a) + bx

Trigonometric Graphs and Radian Measure

• Converting between Degrees and Radians:

Radians=Degrees×π180\text{Radians} = \text{Degrees} \times \frac{\pi}{180}

Degrees=Radians×180π\text{Degrees} = \text{Radians} \times \frac{180}{\pi}

• Arc Length:

s=rθs = r\theta

• Area of a Sector: