How do you solve Interpolation vs. Extrapolation questions on ACT Science graphs and tables?

• Interpolation: Finding a value BETWEEN two existing data points (average the surrounding trends). • Extrapolation: Extending a line or curve BEYOND the plotted range by continuing the established slope/rate of change.

What are the five crucial 'Outside Knowledge' science facts frequently tested on ACT Science without being stated in the passage?

1. pH scale: pH 7 is basic/alkaline. 2. Water phases at 1 atm: freezes at 0°C, boils at 100°C; less dense objects float. 3. Photosynthesis vs. Cellular Respiration: Photosynthesis consumes CO2 + H2O + light -> glucose + O2; Respiration does the reverse. 4. Opposite electrical charges (+ and -) attract; like charges repel. 5. Kinetic Energy (1/2 mv^2) vs. Gravitational Potential Energy (mgh) conservation.

When two parallel lines are cut by a transversal, which angle pairs are congruent and which are supplementary?

• Congruent (equal): Alternate Interior, Alternate Exterior, Corresponding, and Vertical angles. • Supplementary (sum to 180°): Consecutive Interior (Same-Side Interior) angles and Linear Pairs.

What is the relationship between a Radius and a Tangent Line at the point of tangency on a circle?

A tangent line to a circle is ALWAYS perpendicular (forms a 90° right angle) to the radius drawn to the point of tangency. Thus, their slopes are negative reciprocals (m_tangent = -1 / m_radius).

What is the Inscribed Angle Theorem vs. Central Angle in a circle?

A central angle (vertex at the center) equals the measure of its intercepted arc. An inscribed angle (vertex on the circumference) equals HALF the measure of its intercepted arc (θ_inscribed = (1/2) * θ_central). Corollary: Any angle inscribed in a semicircle (intercepting a 180° diameter) is a 90° right angle!

What are the formulas for the Sum of Interior Angles, Each Interior Angle of a regular n-gon, and the Sum of Exterior Angles of any polygon?

• Sum of Interior Angles = (n - 2) * 180°. • Each Interior Angle of a regular n-gon = [(n - 2) * 180°] / n. • Sum of Exterior Angles (one per vertex) of ANY convex polygon = 360° (so each exterior angle of a regular n-gon = 360° / n).

What is the formula for the Area of an Equilateral Triangle of side s and a Regular Hexagon of side s?

• Equilateral Triangle: Area = (s^2 * sqrt(3)) / 4 (altitude h = s*sqrt(3)/2). • Regular Hexagon (6 equilateral triangles): Area = 6 * [(s^2 * sqrt(3)) / 4] = (3 * s^2 * sqrt(3)) / 2.

What is the 3D Space Diagonal (longest interior distance) of a rectangular prism with dimensions l, w, and h?

By the 3D Pythagorean Theorem: d = sqrt(l^2 + w^2 + h^2). For a cube of side s, the space diagonal is s * sqrt(3).

In the sinusoidal function y = A * sin(B(x - C)) + D, how do you find the Amplitude, Period, and Midline?

• Amplitude = |A| = (Max - Min) / 2. • Period = 2π / |B| (for sin/cos; for tan, Period = π / |B|). • Midline (vertical shift) = y = D = (Max + Min) / 2.

How do you distinguish Direct vs. Inverse Relationships in ACT Science multi-variable tables?

Find two trials where only the two variables in question change while others are held constant: • Direct: As Variable A increases, Variable B ALSO increases. • Inverse: As Variable A increases, Variable B DECREASES.

What is the standard equation of an Ellipse on the ACT Math section?

(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1, centered at (h, k), with horizontal semi-axis 'a' and vertical semi-axis 'b'.

In ACT Science Conflicting Viewpoints passages (Scientist 1 vs. Scientist 2), what is the optimal reading strategy?

Unlike graph passages, read the brief intro (to see what phenomenon they agree occurs) and then read the FIRST and LAST sentences of each scientist's viewpoint to lock in their core thesis and mechanism before answering questions comparing them.

On the ACT Math section, how do you calculate the Determinant of a 2x2 Matrix [[a, b], [c, d]] and when is a matrix invertible?

det(A) = ad - bc. A square matrix has an inverse if and only if its determinant is non-zero (ad - bc != 0). Two matrices M_(m x n) and N_(p x q) can be multiplied (M * N) if and only if inner dimensions match (n = p), producing an (m x q) matrix.

On the ACT Math section, what is the cycle of powers of the imaginary unit i = sqrt(-1), and how do you divide complex numbers?

• Cycle of 4: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1 (divide exponent by 4 and use the remainder). • To simplify (a + bi) / (c + di), multiply numerator and denominator by the complex conjugate (c - di); the denominator becomes c^2 + d^2.

How do you convert between Degrees and Radians, and how many radians are in a full circle?

A full 360° circle is 2π radians, so 180° = π radians. • Degrees to Radians: multiply by (π / 180°). • Radians to Degrees: multiply by (180° / π).

In right triangle trigonometry, if tan(θ) = 5/12 for an acute angle θ, how do you quickly find sin(θ) and cos(θ)?

Sketch a right triangle with Opposite = 5 and Adjacent = 12. By the 5-12-13 Pythagorean triple, Hypotenuse = 13. Therefore, sin(θ) = 5/13 and cos(θ) = 12/13.

In ACT Science experiments, what is the purpose of a Control Group (or Blank/Baseline Trial)?

To serve as a baseline standard of comparison in which the independent variable (or treatment) is absent or held at normal conditions, verifying that observed changes in the dependent variable are actually caused by the experimental treatment.

What is the Complementary Angle Identity for Sine and Cosine tested on nearly every Digital SAT?

In any right triangle with acute angles A and B (where A + B = 90° or π/2 radians): sin(A) = cos(B) = cos(90° - A), and cos(A) = sin(B) = sin(90° - A). Why? Because the side Opposite angle A is the exact same side Adjacent to angle B!

What is the Standard Form Equation of a Circle in the xy-plane, and how do you convert x^2 + y^2 + Dx + Ey + F = 0 into it?

(x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius (note: the signs inside parentheses are opposite, and the right side is r SQUARED!). Convert general form by completing the square for both x and y: h = -D/2, k = -E/2, and r^2 = F_moved + h^2 + k^2.

What is the fundamental Pythagorean Trigonometric Identity and how are Law of Sines and Law of Cosines used on the ACT?

• Pythagorean Identity: sin^2(θ) + cos^2(θ) = 1 (holds for all θ). • Law of Sines: sin(A)/a = sin(B)/b = sin(C)/c. • Law of Cosines: c^2 = a^2 + b^2 - 2ab*cos(C) (used for SAS or SSS non-right triangles).

In Similar Triangles (proven by AA similarity), how do the ratios of corresponding lengths, perimeters, areas, and volumes scale?

If the linear scale factor of corresponding sides is k = a / b: • Ratio of Perimeters, Altitudes, or Medians = k • Ratio of Areas (or Surface Areas) = k^2 • Ratio of Volumes (for similar 3D solids) = k^3.

How are sin(θ), cos(θ), and tan(θ) defined in a right triangle (SOH-CAH-TOA) and on the Unit Circle?

• sin(θ) = Opposite / Hypotenuse = y-coordinate on the unit circle. • cos(θ) = Adjacent / Hypotenuse = x-coordinate on the unit circle. • tan(θ) = Opposite / Adjacent = sin(θ) / cos(θ) = y / x.

What are the Volume formulas for a Prism/Cylinder vs. a Pyramid/Cone vs. a Sphere?

• Any Right Prism or Cylinder: V = (Area of Base) * h (for cylinder, V = πr^2h). • Any Pyramid or Cone (comes to a point): V = (1/3) * (Area of Base) * h (for cone, V = (1/3)πr^2h). • Sphere: V = (4/3)πr^3; Surface Area = 4πr^2.

In ACT Science Research Summaries passages, how do you distinguish the Independent Variable, Dependent Variable, and Controlled Variables?

• Independent Variable (IV): The factor deliberately varied by the experimenter across trials (usually on the x-axis or leftmost table column). • Dependent Variable (DV): The outcome measured in response to the IV (usually on the y-axis). • Controlled Variables (Constants): Factors kept identical across all trials to prevent confounding.

What are the side-length ratios of the two Special Right Triangles (45°-45°-90° and 30°-60°-90°)?

• 45°-45°-90° (Isosceles Right Triangle): x : x : x*sqrt(2). • 30°-60°-90° (Half of an Equilateral Triangle): x (opposite 30°) : x*sqrt(3) (opposite 60°) : 2x (hypotenuse).

What are the four most common Pythagorean Triples tested on the SAT and ACT?

• 3 - 4 - 5 (and multiples: 6-8-10, 9-12-15, 15-20-25) • 5 - 12 - 13 (and 10-24-26) • 8 - 15 - 17 • 7 - 24 - 25 Recognizing these saves significant calculation time on right-triangle problems.

What are the formulas for Arc Length s and Sector Area A of a circle in Degrees vs. Radians?

Both equal the fraction of the full circle (θ / 360° or θ / 2π) times the circumference or area: • In Degrees: Arc Length = (θ / 360°) * 2πr; Sector Area = (θ / 360°) * πr^2. • In Radians: Arc Length s = r * θ; Sector Area A = (1/2) * r^2 * θ.

Which trigonometric functions are positive in each of the four Quadrants of the xy-plane (ASTC)?

'All Students Take Calculus': • Quadrant I (0 to π/2): ALL (sin, cos, tan) are positive. • Quadrant II (π/2 to π): Only SINE (y > 0, x y/x > 0) is positive. • Quadrant IV (3π/2 to 2π): Only COSINE (x > 0, y < 0) is positive.

What are the Area formulas for a Trapezoid and a Rhombus/Kite?

• Trapezoid: Area = [(b1 + b2) / 2] * h = (Average of parallel bases) * height. • Rhombus or Kite (perpendicular diagonals d1 and d2): Area = (1/2) * d1 * d2.

In ACT Science Data Representation passages, what is the 'Locators First' strategy?

Skip reading the introductory text! Go straight to the questions and identify the 3 Locators: (1) Which Figure/Table? (2) Which axes/columns and units? (3) What exact value or trend? 80% of Data Representation questions require zero reading of the passage text.

What are the Triangle Inequality Theorem and the Exterior Angle Theorem?

• Triangle Inequality: The third side c of any triangle must lie strictly between the difference and the sum of the other two sides: |a - b| < c < a + b. • Exterior Angle Theorem: An exterior angle of a triangle equals the sum of the two remote (non-adjacent) interior angles.

In ACT Science 'Bridge' (Multi-Step Synthesis) questions, how do you connect Figure 1 and Table 2?

Find the SHARED variable (the 'bridge' column/axis) that appears in both Figure 1 and Table 2. Look up the given input in Figure 1 to read out the bridge variable's value, then take that exact value into Table 2 to find the final answer.

Geometry, Trigonometry, Complex Numbers & ACT Science Data Interpretation

High-yield SAT & ACT geometry, circle equations, radian arc length/sector area, right-triangle and unit-circle trigonometry, matrices, complex numbers, and ACT Science experimental data reasoning.

Level: advanced
Languages: en → en
Cards: 32
Markdown: Geometry, Trigonometry, Complex Numbers & ACT Science Data Interpretation (.md)

Flashcards (32)

#	Front (Term / Question)	Back (Definition / Answer)
1	How do you solve Interpolation vs. Extrapolation questions on ACT Science graphs and tables?	• Interpolation: Finding a value BETWEEN two existing data points (average the surrounding trends). • Extrapolation: Extending a line or curve BEYOND the plotted range by continuing the established slope/rate of change.
2	What are the five crucial 'Outside Knowledge' science facts frequently tested on ACT Science without being stated in the passage?	1. pH scale: pH < 7 is acidic (high [H+]), pH = 7 is neutral, pH > 7 is basic/alkaline. 2. Water phases at 1 atm: freezes at 0°C, boils at 100°C; less dense objects float. 3. Photosynthesis vs. Cellular Respiration: Photosynthesis consumes CO2 + H2O + light -> glucose + O2; Respiration does the reverse. 4. Opposite electrical charges (+ and -) attract; like charges repel. 5. Kinetic Energy (1/2 mv^2) vs. Gravitational Potential Energy (mgh) conservation.
3	When two parallel lines are cut by a transversal, which angle pairs are congruent and which are supplementary?	• Congruent (equal): Alternate Interior, Alternate Exterior, Corresponding, and Vertical angles. • Supplementary (sum to 180°): Consecutive Interior (Same-Side Interior) angles and Linear Pairs.
4	What is the relationship between a Radius and a Tangent Line at the point of tangency on a circle?	A tangent line to a circle is ALWAYS perpendicular (forms a 90° right angle) to the radius drawn to the point of tangency. Thus, their slopes are negative reciprocals (m_tangent = -1 / m_radius).
5	What is the Inscribed Angle Theorem vs. Central Angle in a circle?	A central angle (vertex at the center) equals the measure of its intercepted arc. An inscribed angle (vertex on the circumference) equals HALF the measure of its intercepted arc (θ_inscribed = (1/2) * θ_central). Corollary: Any angle inscribed in a semicircle (intercepting a 180° diameter) is a 90° right angle!
6	What are the formulas for the Sum of Interior Angles, Each Interior Angle of a regular n-gon, and the Sum of Exterior Angles of any polygon?	• Sum of Interior Angles = (n - 2) * 180°. • Each Interior Angle of a regular n-gon = [(n - 2) * 180°] / n. • Sum of Exterior Angles (one per vertex) of ANY convex polygon = 360° (so each exterior angle of a regular n-gon = 360° / n).
7	What is the formula for the Area of an Equilateral Triangle of side s and a Regular Hexagon of side s?	• Equilateral Triangle: Area = (s^2 * sqrt(3)) / 4 (altitude h = ssqrt(3)/2). • Regular Hexagon (6 equilateral triangles): Area = 6 [(s^2 * sqrt(3)) / 4] = (3 * s^2 * sqrt(3)) / 2.
8	What is the 3D Space Diagonal (longest interior distance) of a rectangular prism with dimensions l, w, and h?	By the 3D Pythagorean Theorem: d = sqrt(l^2 + w^2 + h^2). For a cube of side s, the space diagonal is s * sqrt(3).
9	In the sinusoidal function y = A * sin(B(x - C)) + D, how do you find the Amplitude, Period, and Midline?	• Amplitude = \|A\| = (Max - Min) / 2. • Period = 2π / \|B\| (for sin/cos; for tan, Period = π / \|B\|). • Midline (vertical shift) = y = D = (Max + Min) / 2.
10	How do you distinguish Direct vs. Inverse Relationships in ACT Science multi-variable tables?	Find two trials where only the two variables in question change while others are held constant: • Direct: As Variable A increases, Variable B ALSO increases. • Inverse: As Variable A increases, Variable B DECREASES.
11	What is the standard equation of an Ellipse on the ACT Math section?	(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1, centered at (h, k), with horizontal semi-axis 'a' and vertical semi-axis 'b'.
12	In ACT Science Conflicting Viewpoints passages (Scientist 1 vs. Scientist 2), what is the optimal reading strategy?	Unlike graph passages, read the brief intro (to see what phenomenon they agree occurs) and then read the FIRST and LAST sentences of each scientist's viewpoint to lock in their core thesis and mechanism before answering questions comparing them.
13	On the ACT Math section, how do you calculate the Determinant of a 2x2 Matrix [[a, b], [c, d]] and when is a matrix invertible?	det(A) = ad - bc. A square matrix has an inverse if and only if its determinant is non-zero (ad - bc != 0). Two matrices M_(m x n) and N_(p x q) can be multiplied (M * N) if and only if inner dimensions match (n = p), producing an (m x q) matrix.
14	On the ACT Math section, what is the cycle of powers of the imaginary unit i = sqrt(-1), and how do you divide complex numbers?	• Cycle of 4: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1 (divide exponent by 4 and use the remainder). • To simplify (a + bi) / (c + di), multiply numerator and denominator by the complex conjugate (c - di); the denominator becomes c^2 + d^2.
15	How do you convert between Degrees and Radians, and how many radians are in a full circle?	A full 360° circle is 2π radians, so 180° = π radians. • Degrees to Radians: multiply by (π / 180°). • Radians to Degrees: multiply by (180° / π).
16	In right triangle trigonometry, if tan(θ) = 5/12 for an acute angle θ, how do you quickly find sin(θ) and cos(θ)?	Sketch a right triangle with Opposite = 5 and Adjacent = 12. By the 5-12-13 Pythagorean triple, Hypotenuse = 13. Therefore, sin(θ) = 5/13 and cos(θ) = 12/13.
17	In ACT Science experiments, what is the purpose of a Control Group (or Blank/Baseline Trial)?	To serve as a baseline standard of comparison in which the independent variable (or treatment) is absent or held at normal conditions, verifying that observed changes in the dependent variable are actually caused by the experimental treatment.
18	What is the Complementary Angle Identity for Sine and Cosine tested on nearly every Digital SAT?	In any right triangle with acute angles A and B (where A + B = 90° or π/2 radians): sin(A) = cos(B) = cos(90° - A), and cos(A) = sin(B) = sin(90° - A). Why? Because the side Opposite angle A is the exact same side Adjacent to angle B!
19	What is the Standard Form Equation of a Circle in the xy-plane, and how do you convert x^2 + y^2 + Dx + Ey + F = 0 into it?	(x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius (note: the signs inside parentheses are opposite, and the right side is r SQUARED!). Convert general form by completing the square for both x and y: h = -D/2, k = -E/2, and r^2 = F_moved + h^2 + k^2.
20	What is the fundamental Pythagorean Trigonometric Identity and how are Law of Sines and Law of Cosines used on the ACT?	• Pythagorean Identity: sin^2(θ) + cos^2(θ) = 1 (holds for all θ). • Law of Sines: sin(A)/a = sin(B)/b = sin(C)/c. • Law of Cosines: c^2 = a^2 + b^2 - 2ab*cos(C) (used for SAS or SSS non-right triangles).
21	In Similar Triangles (proven by AA similarity), how do the ratios of corresponding lengths, perimeters, areas, and volumes scale?	If the linear scale factor of corresponding sides is k = a / b: • Ratio of Perimeters, Altitudes, or Medians = k • Ratio of Areas (or Surface Areas) = k^2 • Ratio of Volumes (for similar 3D solids) = k^3.
22	How are sin(θ), cos(θ), and tan(θ) defined in a right triangle (SOH-CAH-TOA) and on the Unit Circle?	• sin(θ) = Opposite / Hypotenuse = y-coordinate on the unit circle. • cos(θ) = Adjacent / Hypotenuse = x-coordinate on the unit circle. • tan(θ) = Opposite / Adjacent = sin(θ) / cos(θ) = y / x.
23	What are the Volume formulas for a Prism/Cylinder vs. a Pyramid/Cone vs. a Sphere?	• Any Right Prism or Cylinder: V = (Area of Base) * h (for cylinder, V = πr^2h). • Any Pyramid or Cone (comes to a point): V = (1/3) * (Area of Base) * h (for cone, V = (1/3)πr^2h). • Sphere: V = (4/3)πr^3; Surface Area = 4πr^2.
24	In ACT Science Research Summaries passages, how do you distinguish the Independent Variable, Dependent Variable, and Controlled Variables?	• Independent Variable (IV): The factor deliberately varied by the experimenter across trials (usually on the x-axis or leftmost table column). • Dependent Variable (DV): The outcome measured in response to the IV (usually on the y-axis). • Controlled Variables (Constants): Factors kept identical across all trials to prevent confounding.
25	What are the side-length ratios of the two Special Right Triangles (45°-45°-90° and 30°-60°-90°)?	• 45°-45°-90° (Isosceles Right Triangle): x : x : xsqrt(2). • 30°-60°-90° (Half of an Equilateral Triangle): x (opposite 30°) : xsqrt(3) (opposite 60°) : 2x (hypotenuse).
26	What are the four most common Pythagorean Triples tested on the SAT and ACT?	• 3 - 4 - 5 (and multiples: 6-8-10, 9-12-15, 15-20-25) • 5 - 12 - 13 (and 10-24-26) • 8 - 15 - 17 • 7 - 24 - 25 Recognizing these saves significant calculation time on right-triangle problems.
27	What are the formulas for Arc Length s and Sector Area A of a circle in Degrees vs. Radians?	Both equal the fraction of the full circle (θ / 360° or θ / 2π) times the circumference or area: • In Degrees: Arc Length = (θ / 360°) * 2πr; Sector Area = (θ / 360°) * πr^2. • In Radians: Arc Length s = r * θ; Sector Area A = (1/2) * r^2 * θ.
28	Which trigonometric functions are positive in each of the four Quadrants of the xy-plane (ASTC)?	'All Students Take Calculus': • Quadrant I (0 to π/2): ALL (sin, cos, tan) are positive. • Quadrant II (π/2 to π): Only SINE (y > 0, x < 0) is positive. • Quadrant III (π to 3π/2): Only TANGENT (y < 0, x < 0 => y/x > 0) is positive. • Quadrant IV (3π/2 to 2π): Only COSINE (x > 0, y < 0) is positive.
29	What are the Area formulas for a Trapezoid and a Rhombus/Kite?	• Trapezoid: Area = [(b1 + b2) / 2] * h = (Average of parallel bases) * height. • Rhombus or Kite (perpendicular diagonals d1 and d2): Area = (1/2) * d1 * d2.
30	In ACT Science Data Representation passages, what is the 'Locators First' strategy?	Skip reading the introductory text! Go straight to the questions and identify the 3 Locators: (1) Which Figure/Table? (2) Which axes/columns and units? (3) What exact value or trend? 80% of Data Representation questions require zero reading of the passage text.
31	What are the Triangle Inequality Theorem and the Exterior Angle Theorem?	• Triangle Inequality: The third side c of any triangle must lie strictly between the difference and the sum of the other two sides: \|a - b\| < c < a + b. • Exterior Angle Theorem: An exterior angle of a triangle equals the sum of the two remote (non-adjacent) interior angles.
32	In ACT Science 'Bridge' (Multi-Step Synthesis) questions, how do you connect Figure 1 and Table 2?	Find the SHARED variable (the 'bridge' column/axis) that appears in both Figure 1 and Table 2. Look up the given input in Figure 1 to read out the bridge variable's value, then take that exact value into Table 2 to find the final answer.