If f(x) = 2x^2 – 3x + 5, find the value of x when f(x) equals 12.To find the value of x when f(x) = 12, solve the equation 2x^2 – 3x + 5 = 12. Simplifying, we get 2x^2 – 3x – 7 = 0. Using the quadratic formula, x = (3 ± √(9 + 56))/4, yielding x = 2 or x = -7/2.
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How do you derive the double angle formulas for sine, cosine, and tangent and apply them to solve complex trigonometric equations?
How do you derive the double angle formulas for sine, cosine, and tangent and apply them to solve complex trigonometric equations?To derive the double angle formulas for sine, cosine, and tangent, use the identities sin(2θ) = 2sin(θ)cos(θ), cos(2θ) = cos²(θ) – sin²(θ), and tan(2θ) = 2tan(θ)/(1-tan²(θ)). Apply these formulas by substituting them into trigonometric equations to simplify and solve for unknown angles.
How do you find the sine, cosine, and tangent of an angle in a right triangle using the Pythagorean Theorem?
How do you find the sine, cosine, and tangent of an angle in a right triangle using the Pythagorean Theorem?To find the sine, cosine, and tangent of an angle in a right triangle using the Pythagorean Theorem, first identify the lengths of the sides: opposite (a), adjacent (b), and hypotenuse (c). The Pythagorean Theorem states that a² + b² = c². Then, sine is a/c, cosine is b/c, and tangent is a/b.
Can you explain how to use the sum and difference formulas for sine, cosine, and tangent to solve complex trigonometric expressions?
Can you explain how to use the sum and difference formulas for sine, cosine, and tangent to solve complex trigonometric expressions?The sum and difference formulas for sine, cosine, and tangent are vital tools in trigonometry. They allow us to simplify complex expressions by breaking them into manageable parts. For sine: sin(a ± b) = sin(a)cos(b) ± cos(a)sin(b). For cosine: cos(a ± b) = cos(a)cos(b) ∓ sin(a)sin(b). For tangent: tan(a ± b) = (tan(a) ± tan(b)) / (1 ∓ tan(a)tan(b)). These formulas are particularly useful in solving equations, proving identities, and evaluating trigonometric functions at specific angles.
How do you find the angle of a right triangle using inverse trigonometric functions given the lengths of two sides?
How do you find the angle of a right triangle using inverse trigonometric functions given the lengths of two sides?To find an angle in a right triangle using inverse trigonometric functions, use the ratios of the side lengths. For angle θ, if you know the opposite side (a) and the adjacent side (b), use θ = arctan(a/b). If you know the hypotenuse (c) and the opposite side (a), use θ = arcsin(a/c). For the hypotenuse (c) and the adjacent side (b), use θ = arccos(b/c).
How do you simplify the expression with both multi-step operations and fraction coefficients: 2/3x – 5/6 = 1/2( x + 10 )?
How do you simplify the expression with both multi-step operations and fraction coefficients: 2/3x – 5/6 = 1/2( x + 10 )?To simplify the expression 2/3x – 5/6 = 1/2(x + 10), first distribute the 1/2 on the right side: 2/3x – 5/6 = 1/2x + 5. Then, to eliminate the fractions, multiply every term by 6: 4x – 5 = 3x + 30. Finally, solve for x by isolating it: x = 35.
How do you derive the formula for the foci of an ellipse given the lengths of the major and minor axes?
How do you derive the formula for the foci of an ellipse given the lengths of the major and minor axes?To derive the formula for the foci of an ellipse, start with the standard form of the ellipse equation: (x^2/a^2) + (y^2/b^2) = 1, where a is the semi-major axis and b is the semi-minor axis. The distance from the center to each focus (c) is given by the formula c = √(a^2 – b^2).
What is the difference between an integer and a whole number?
What is the difference between an integer and a whole number?An integer is any number from the set of positive and negative whole numbers, including zero (e.g., -3, 0, 4). Whole numbers are a subset of integers that include only non-negative numbers (e.g., 0, 1, 2, 3). Thus, all whole numbers are integers, but not all integers are whole numbers.
How do you differentiate between a population and a sample in statistics, and why is it important to understand the difference?
How do you differentiate between a population and a sample in statistics, and why is it important to understand the difference?In statistics, a population refers to the entire group about which data is being collected, while a sample is a subset of the population. Understanding the difference is crucial because it affects the accuracy and generalizability of statistical inferences. Sampling allows for manageable data collection and analysis, but it also introduces sampling error.
How do you derive the equation of an ellipse given its major and minor axes lengths as well as its foci coordinates, and prove that the distance sum of a point to its foci is constant?
How do you derive the equation of an ellipse given its major and minor axes lengths as well as its foci coordinates, and prove that the distance sum of a point to its foci is constant?Given the lengths of the major axis (2a) and minor axis (2b), and the coordinates of the foci at (±c, 0), the standard form of the ellipse equation is (x²/a²) + (y²/b²) = 1. To prove the distance sum is constant, note that for any point (x, y) on the ellipse, the sum of distances to the foci (d1 + d2) equals 2a, which is constant.
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