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00:00:00 – 00:11:53
The video is a detailed review of essential math formulas and concepts crucial for SAT preparation, focusing on their practical applications. The presenter covers a variety of key formulas including those for geometry, algebra, and trigonometry. Important formulas discussed include the slope and point-slope forms for linear equations, the distance and midpoint formulas, and quadratic-related equations such as the quadratic formula and the discriminant. The video also explains circle-related formulas, trigonometry basics (including SOHCAHTOA and the relationships between sine and cosine), as well as exponential growth and decay. Additionally, there is a discussion on Pythagorean theorem and common Pythagorean triples, essential properties of polygons, and strategies like finding the sum of solutions for quadratics. Emphasizing practical SAT problem-solving, the video illustrates these formulas with examples and highlights the importance of memorizing them for exam success.
00:00:00
In this part of the video, the presenter provides a comprehensive review of essential SAT math formulas. He emphasizes the importance of these formulas and points out that certain area and volume formulas are already provided on the SAT formula sheet. Key formulas discussed include:
1. **Slope Formula**: Used for calculating the slope of a line, given by ((y1 – y2) / (x1 – x2)).
2. **Slope-Intercept Form**: Expressed as (y = mx + b), where (m) is the slope and (b) is the y-intercept.
3. **Point-Slope Form**: Given by (y – y1 = m(x – x1)), where (m) is the slope and ((x1, y1)) is a point on the line.
4. **Midpoint Formula**: Calculates the midpoint of a line segment, given by ((x1 + x2) / 2, (y1 + y2) / 2).
5. **Distance Formula**: Determines the distance between two points, using (sqrt{(x1 – x2)^2 + (y1 – y2)^2}).
Each formula is illustrated with examples to demonstrate their application.
00:03:00
In this part of the video, the presenter covers various mathematical formulas and their applications.
1. **Arc Length of a Circle**: The formula provided is ((n/360) times 2pi r), where (n) is the central angle in degrees.
2. **Area of a Sector**: The formula is ((n/360) times pi r^2), with (n) being the central angle in degrees.
3. **Quadratic Formula**: Explained as (-b pm sqrt{b^2 – 4ac} / 2a) for finding the roots of quadratics.
4. **SOHCAHTOA (Trigonometry)**: Definitions of sine (opposite/hypotenuse), cosine (adjacent/hypotenuse), and tangent (opposite/adjacent) related to a given angle.
5. **Probability**: Explained as the ratio of favorable outcomes to total outcomes, exemplified by selecting red marbles from a bag.
6. **Circle Equation**: Detailed as ((x-h)^2 + (y-k)^2 = r^2), with ((h, k)) being the center and (r) the radius.
7. **Exponential Growth and Decay**: The formula is (y = a(1 pm r)^t), where (a) is the initial value, (r) is the growth rate, and (t) is time.
The segment concludes with a quick example calculation for exponential growth.
00:06:00
In this segment, the video covers several key mathematical concepts. It begins with an explanation of exponential decay, detailing how to adjust calculations for a decay rate, illustrated with a 150 initial value decreasing by 9% over two years. Next, it moves to finding the vertex of a parabola in standard form using the formula -b/2a and explains how to convert this to vertex form, noting the vertex is located at (h, k). The Pythagorean theorem is also reviewed, with a squared plus b squared equaling c squared, emphasizing the parts of a right triangle. The distance formula, d = rt, is discussed alongside a reminder to use consistent units for rate and time.
It then introduces the relationship between sine and cosine in trigonometry, stating that sine of an angle equals the cosine of its complement (90 degrees minus the angle). This is followed by a method to find the sum of solutions for a quadratic in standard form, given by -b/a, illustrated with an example equation. Finally, the discriminant of a quadratic equation, b squared minus 4ac, is addressed, noting its role in the quadratic formula and its importance for determining the nature of the solutions (real or non-real).
00:09:00
In this segment of the video, the speaker covers several mathematical concepts and formulas important for the SAT. First, they discuss the discriminant and its implications: a zero discriminant means one real solution, while a positive discriminant means two real solutions. Next, the area formula for an equilateral triangle is explained with examples using side lengths of 1, 2, and 3. Following this, Pythagorean triples are introduced, emphasizing their usefulness in saving time during the SAT. Common triples such as 3-4-5, 5-12-13, 7-24-25, and 8-15-17 are mentioned, along with their multiples. Then, the formula for calculating the perpendicular slope of a line is provided – it’s the negative reciprocal of the original slope. Lastly, the formula for the sum of interior angles of a polygon is explained, with examples given for triangles, quadrilaterals, and pentagons. The speaker advises viewers to memorize these formulas for SAT success.