Comprehensive Study Notes: Mathematical Functions, Geometry, and Trigonometry

Factoring Techniques

• Greatest Common Factor (GCF) and Difference of Two Squares (DOTS):

  • Example 1: $100m^2 - 16$

    • Step 1 (GCF): Factor out $4$ to get $4(25n^2 - 4)$.

    • Step 2 (DOTS): $25n^2$ and $4$ are perfect squares. Factor as $4(5n + 2)(5n - 2)$.

• Trinomial Factoring ($a = 1$):

  • Example 2: $4n^2 + 32n + 64$

    • Step 1 (GCF): Factor out $4$ to get $4(n^2 + 8n + 16)$.

    • Step 2 (Trinomial): Look for numbers that multiply to $16$ and add to $8$. Factors as $4(n + 4)(n + 4)$ or $4(n + 4)^2$.

• Factoring by Grouping:

  • Example 3: $18k^4 + 48k^2 + 32$

    • Step 1 (GCF): Factor out $2$ to get $2(9k^4 + 24k^2 + 16)$.

    • Step 2 (Grouping): Split the middle term: $2((9k^4 + 12k^2) + (12k^2 + 16))$.

    • Step 3: Factor each group: $2(3k^2(3k^2 + 4) + 4(3k^2 + 4))$.

    • Step 4: Final form: $2(3k^2 + 4)(3k^2 + 4) = 2(3k^2 + 4)^2$.

  • Example 6: $5p^3 + 10p^2 - 7p - 14$

    • Group into pairs: $(5p^3 + 10p^2) - (7p + 14)$.

    • Factor: $5p^2(p + 2) - 7(p + 2)$.

    • Result: $(5p^2 - 7)(p + 2)$.

• Sum and Difference of Cubes:

  • Formulas used: $(a^3 + b^3) = (a + b)(a^2 - ab + b^2)$ and $(a^3 - b^3) = (a - b)(a^2 + ab + b^2)$.

  • Example 4 (Sum of Cubes): $27x^3 + 125$

    • $a = 3x$, $b = 5$.

    • Steps: Square the front ($9x^2$), multiply them ($15x$), square the back ($25$). Change the sign of the middle term.

    • Result: $(3x + 5)(9x^2 - 15x + 25)$.

  • Example 5 (Difference of Cubes): $8x^3 - 27y^3$

    • $a = 2x$, $b = 3y$.

    • Result: $(2x - 3y)(4x^2 + 6xy + 9y^2)$.

• Factoring with Negative and Fractional Exponents:

  • Example 7: $6x^{3/2} + x^{1/2} - 2x^{-1/2}$

    • Step 1: Factor out the variable with the smallest exponent: $x^{-1/2}$.

    • Step 2: $x^{-1/2}(6x^2 + x - 2)$.

    • Step 3: Factor the trinomial by decomposition: $6x^2 + 4x - 3x - 2$.

    • Step 4: $2x(3x + 2) - 1(3x + 2)$.

    • Final result: $x^{-1/2}(2x - 1)(3x + 2)$.

Quadratic Inequalities

• Example 8: Solve 5x^2 - 2x - 3 > 0.

  • Step 1: Find critical values by setting the expression to zero: $5x^2 - 5x + 3x - 3 = 0$.

  • Step 2: Factor as $5x(x - 1) + 3(x - 1) = 0$, leading to $(5x + 3)(x - 1) = 0$.

  • Critical values: $x = -\frac{3}{5}$ and $x = 1$.

  • Step 3: Test intervals. The inequality is satisfied when x < -\frac{3}{5} or x > 1.

  • Interval Notation: $(-\text{inf}, -\frac{3}{5}) \text{ U } (1, \text{inf})$.

Operations with Radicals and Complex Numbers

• Simplifying Radicals:

  • Example 9a: $\sqrt{28x^3y^5}$

    • $\sqrt{4x^2y^4} \times \sqrt{7xy} = 2xy^2\sqrt{7xy}$.

  • Example 9b: $\sqrt[3]{135a^8b^7}$

    • $\sqrt[3]{27a^6b^6} \times \sqrt[3]{5a^2b} = 3a^2b^2 \sqrt[3]{5a^2b}$.

  • Example 9c: $\sqrt[4]{500x^5y^6}$

    • $\sqrt[4]{625}$ is not a factor; using $125$ for cube roots but here it is a 4th root problem. Correction: $\sqrt[4]{625x^4y^4}$ is $5xy$, but for $500$, simplification depends on perfect 4th powers ($16, 81, 256, 625$). The transcript shows: $20xy \sqrt[4]{x^2}$.

• Complex Numbers in $a + bi$ form:

  • Simplifying expressions:

    • $-\sqrt{64} + \sqrt{-160} - \sqrt{-4} - \sqrt{-100}$

    • $-8 + 4i\sqrt{10} - 2i - 10i$

    • Result: $-8 - 12i + 4i\sqrt{10} = -8 - 12i + 4\sqrt{10}i$

  • Multiplying radicals with imaginaries:

    • $(4\sqrt{-20})(-6i - 6)$

    • $(4(2i\sqrt{5}))(-6i - 6) = (8i\sqrt{5})(-6i - 6)$

    • $-48i^2\sqrt{5} - 48i\sqrt{5}$

    • Since $i^2 = -1$, result is $48\sqrt{5} - 48\sqrt{5}i$.

Sum and Product of Roots

• Finding the Quadratic Equation:

  • Formula: $x^2 - (\text{sum})x + (\text{product}) = 0$.

  • Relationships: $\text{Sum} = -\frac{b}{a}$ and $\text{Product} = \frac{c}{a}$.

  • Example 10: Sum $= -3$, Product $= -9$.

    • Equation: $x^2 + 3x - 9 = 0$.

    • Scaling the equation: $2x^2 + 6x - 18 = 0$ also meets these requirements (Option 3).

Geometry: Lines and Segments

• Angle Bisectors:

  • A ray that bisects an angle cuts it into two equal pieces.

  • Example 12: Ray $EF$ bisects $\angle DEG$. If $m\angle FEG = 6x - 7$ and $m\angle FED = 2x + 41$, find $m\angle FEG$.

    • Set equal: $2x + 41 = 6x - 7 \rightarrow 48 = 4x \rightarrow x = 12$.

    • $m\angle FEG = 6(12) - 7 = 65^\circ$.

• Midpoint and Distance Formulas:

  • Midpoint: $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$.

    • Example 13: Endpoints $(2, -5)$ and $(8, 3)$. Midpoint is $(\frac{2+8}{2}, \frac{-5+3}{2}) = (5, -1)$.

  • Distance: $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

    • Example 14: $R(-2, 3)$ and $S(4, 5)$.

    • $D = \sqrt{(4 - (-2))^2 + (5 - 3)^2} = \sqrt{6^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10}$.

• Equations of Perpendicular Lines:

  • Perpendicular lines have negative reciprocal slopes.

  • Example 15: Line through $(3, 3)$ perpendicular to $y = -\frac{1}{2}x + 2$.

    • Original slope $m = -\frac{1}{2}$; Perpendicular slope $m_{\perp} = 2$.

    • Equation: $y - 3 = 2(x - 3)$ or $y = 2x - 3$.

  • Perpendicular Bisector:

    1. Find midpoint of segment.

    2. Find slope of original segment.

    3. Use negative reciprocal slope with the midpoint.

    • Example 16: Points $(0, 5)$ and $(8, 9)$.

    • Midpoint: $(4, 7)$.

    • Slope: $\frac{9 - 5}{8 - 0} = \frac{1}{2}$; $m_{\perp} = -2$.

    • Equation: $y - 7 = -2(x - 4)$ or $y = -2x + 15$.

Transversals, Transformations, and Triangles

• Parallel Lines and Transversals:

  • Theorem: Same-Side Interior angles are supplementary ($180^\circ$).

  • Example 17: $m\angle MTU = 3x - 18$ and $m\angle TUN = 4x - 12$.

    • $3x - 18 + 4x - 12 = 180 \rightarrow 7x - 30 = 180 \rightarrow 7x = 210 \rightarrow x = 30$.

    • $m\angle TUN = 4(30) - 12 = 108^\circ$.

    • $m\angle MTS = 108^\circ$ because corresponding angles are congruent.

• Midsegments and Trapezoids:

  • Example 18: Triangle $ABC$ with midpoints $L, M, N$.

    • If $MN = 8$, then $BC = 16$ (base is twice the midsegment). Since $L$ is the midpoint, $BL = LC = 8$.

    • Side lengths are given as $MN = 8$, $ML = 5$, $NL = 6$.

    • Trapezoid $BMNC$ perimeter: $MN + NC + BC + BM = 8 + 6 + 16 + 5 = 35$.

• Transformations:

  • Reflections: Reflection over the y-axis changes $(x, y)$ to $(-x, y)$.

    • Example 19: Point $(2, -3)$ reflected across y-axis results in $(-2, -3)$.

  • Dilations: Multiply both coordinates by the scale factor.

    • Example 20: Dilation by $1/3$ of $(-12, -9)$ centered at origin results in $(-4, -3)$.

Circle Geometry

• Angles in Circles:

  • Inscribed Angle: Equal to half the measure of its intercepted arc.

    • Example 21: Major arc $ABC = 268^\circ$. Total circle is $360^\circ$. Remaining arc $AC = 360 - 268 = 92^\circ$.

    • $m\angle ABC = \frac{92}{2} = 46^\circ$.

  • Central Angle: Equal to the intercepted arc ($m\angle AOB = mAB$).

  • Angle Vertex Outside Circle: $\frac{\text{Big Arc} - \text{Little Arc}}{2}$.

    • Example 23: Arc $AB = 117^\circ$, Arc $ACB = 243^\circ$.

    • $x = \frac{243 - 117}{2} = \frac{126}{2} = 63^\circ$.

• Equations of Circles (Standard Form):

  • Completing the square to find center $(h, k)$ and radius $r$.

  • Example 22a: $x^2 + 8x + y^2 - 4y = -16$

    • $(x^2 + 8x + 16) + (y^2 - 4y + 4) = -16 + 16 + 4$

    • $(x + 4)^2 + (y - 2)^2 = 4$

    • Center: $(-4, 2)$, Radius: $2$.

  • Example 22b: $x^2 + y^2 - 10y = 75$

    • $x^2 + (y^2 - 10y + 25) = 75 + 25$

    • $x^2 + (y - 5)^2 = 100$

    • Center: $(0, 5)$, Radius: $10$.

Trigonometry Fundamentals

• Ratios (SOHCAHTOA and Reciprocals):

  • $\sin(A) = \frac{\text{opp}}{\text{hyp}}$, $\cos(A) = \frac{\text{adj}}{\text{hyp}}$, $\tan(A) = \frac{\text{opp}}{\text{adj}}$.

  • $\csc(A) = \frac{1}{\sin(A)}$, $\sec(A) = \frac{1}{\cos(A)}$, $\cot(A) = \frac{1}{\tan(A)}$.

  • Example 24: Triangle with side $21$, $20$, hypotenuse $29$.

    • $\sin(A) = \frac{21}{29}$, $\cos(A) = \frac{20}{29}$, $\tan(A) = \frac{21}{20}$.

    • $\csc(A) = \frac{29}{21}$, $\sec(A) = \frac{29}{20}$, $\cot(A) = \frac{20}{21}$.

• Special Right Triangles:

  • $45-45-90$ Triangle: Side ratios are $x, x, x\sqrt{2}$.

  • $30-60-90$ Triangle: Side ratios are $x, x\sqrt{3}, 2x$.

  • Example 25a: Given side $9$ opposite $45^\circ$. Other side is $9$, hypotenuse is $9\sqrt{2}$.

  • Example 25c: Given hypotenuse $\sqrt{5}$ in a $30-60-90$. Short side $x = \frac{\sqrt{5}}{2}$, long side $x\sqrt{3} = \frac{\sqrt{15}}{2}$.

Law of Sines and Ambiguous Case (SSA)

• Determining number of possible triangles:

  • Example 26a: $m\angle C = 63^\circ, b = 9, c = 12$.

    • $\frac{9}{\sin(B)} = \frac{12}{\sin(63)} \rightarrow \sin(B) = \frac{9\sin(63)}{12} \approx 0.668$.

    • $B \approx 41.9^\circ$ or $138.1^\circ$.

    • 138.1 + 63 > 180, so only one triangle exists.

  • Example 26b: $m\angle B = 35^\circ, a = 24, b = 6$.

    • $\frac{24}{\sin(A)} = \frac{6}{\sin(35)} \rightarrow \sin(A) = \frac{24\sin(35)}{6} \approx 2.29$.

    • \sin(A) > 1 is impossible. No triangles exist.

  • Example 26c: $m\angle A = 29^\circ, c = 18, a = 17$.

    • $\sin(C) = \frac{18\sin(29)}{17} \approx 0.513$.

    • $C \approx 30.9^\circ$ or $149.1^\circ$.

    • Both 30.9 + 29 < 180 and 149.1 + 29 < 180. Two triangles exist.

Advanced Equation Solving

• Fractional Exponent Equations:

  • Example 28a: $3 - 54 = -10 - (m - 10)^{3/2}$

    • $-64 = -(m - 10)^{3/2} \rightarrow 64 = (m - 10)^{3/2}$.

    • Raise to the reciprocal power $2/3$: $64^{2/3} = m - 10 \rightarrow 16 = m - 10 \rightarrow m = 26$.

  • Example 28b: $-5126 = -5 - 5(3x + 22)^{5/3}$.

    • $-5120 = -5(3x + 22)^{5/3} \rightarrow 1024 = (3x + 22)^{5/3}$.

    • Raise to power $3/5$: $1024^{3/5} = 3x + 22 \rightarrow 64 = 3x + 22 \rightarrow x = 14$.

• Radical Equations:

  • Example 28c: Solve $\sqrt{2k+40} = -16 - 2k$.

    • Square both sides: $2k + 40 = (-16 - 2k)^2$. Check for extraneous roots.

    • Result found: $k = -14$.

  • Example 28d: $\sqrt{x+8} = \sqrt{3x+8}$.

    • $x + 8 = 3x + 8 \rightarrow 0 = 2x \rightarrow x = 0$.

• Absolute Value Inequalities:

  • Theorem: |X| < a means -a < X < a.

  • Example 28e: 9|m - 8| - 10 < 26

    • 9|m - 8| < 36 \rightarrow |m - 8| < 4.

    • -4 < m - 8 < 4 \rightarrow 4 < m < 12. Interval: $(4, 12)$.

  • Example 28f: 9|x + 8| + 10 < 55

    • 9|x + 8| < 45 \rightarrow |x + 8| < 5.

    • -5 < x + 8 < 5 \rightarrow -13 < x < -3. Interval: $(-13, -3)$.

Mathematical Proofs and Geometry Logic

• Quadrants for Trig Functions:

  • ASTC (All, Sine, Tangent, Cosine) identifies positivity in quadrants I, II, III, IV respectively.

  • Example 29: If \sin(\theta) > 0 (Quadrant I or II) and \sec(\theta) < 0 (Quadrant II or III), the terminal side lies in Quadrant II.

• Parallelogram Properties:

  1. Opposite sides are parallel and congruent.

  2. Opposite angles are congruent.

  3. Consecutive angles are supplementary.

  • Example 30a: If one angle is $78^\circ$, the opposite angle $z = 102^\circ$ (Wait, consecutive angles are supplementary, so $180 - 78 = 102$). Alternate interior angles are equal: $x = 29^\circ, y = 73^\circ$.

  • Proof of non-parallelogram: If opposite sides are not equal (e.g., $10 \neq 8$) or opposite angles are not equal (e.g., $135 \neq 130$), it is not a parallelogram.

• Geometric Proof (ASA):

  • Given: $\angle JKL ≅ \angle MLK$ and $\angle JLK ≅ \angle MKL$.

  • Goal: Prove $JK ≅ ML$.

  • Steps:

    1. $\angle JKL ≅ \angle MLK, \angle JLK ≅ \angle MKL$ (Given).

    2. $KL ≅ KL$ (Reflexive Property).

    3. $\Delta JKL ≅ \Delta MLK$ (Angle-Side-Angle or ASA).

    4. $JK ≅ ML$ (Corresponding Parts of Congruent Triangles are Congruent or CPCTC).

Unit Circle and Quadratic Roots

• Unit Circle Coordinates (x, y) = (̄\cos ̄\theta, ̄\sin ̄\theta).

  • $30^\circ (π/6)$: $(\frac{\sqrt{3}}{2}, \frac{1}{2})$

  • $45^\circ (π/4)$: $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$

  • $60^\circ (π/3)$: $(\frac{1}{2}, \frac{\sqrt{3}}{2})$

  • $90^\circ (π/2)$: $(0, 1)$

  • $180^\circ (π)$: $(-1, 0)$

  • $270^\circ (3π/2)$: $(0, -1)$

• Quadratic Formula Roots:

  • Example 33: $x^2 - 3x + 10 = 0$.

    • $x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(10)}}{2(1)}$.

    • $x = \frac{3 \pm \sqrt{-31}}{2} = \frac{3 \pm i\sqrt{31}}{2}$.

• Radians and Degrees:

  • Multiply by $\frac{180}{\pi}$.

  • Example 32: $\frac{19\pi}{18} = \frac{19 \times 180}{18} = 190^\circ$ (Quadrant III).