What is the difference between sine, cosine, and tangent in trigonometry?Sine, cosine, and tangent are fundamental trigonometric functions that relate the angles of a right triangle to the lengths of its sides. Sine (sin) is the ratio of the opposite side to the hypotenuse. Cosine (cos) is the ratio of the adjacent side to the hypotenuse. Tangent (tan) is the ratio of the opposite side to the adjacent side.
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What is the difference between mean, median, and mode in statistics?
What is the difference between mean, median, and mode in statistics?In statistics, the mean is the average of a data set, calculated by summing all values and dividing by the number of values. The median is the middle value when the data set is ordered, providing a measure of central tendency that is less affected by outliers. The mode is the most frequently occurring value in a data set, representing the value with the highest frequency.
How do you calculate the interquartile range (IQR) of a data set and why is it important in determining data variability?
How do you calculate the interquartile range (IQR) of a data set and why is it important in determining data variability?To calculate the interquartile range (IQR) of a data set, first arrange the data in ascending order. Divide the data into four equal parts to find the first quartile (Q1) and the third quartile (Q3). The IQR is the difference between Q3 and Q1 (IQR = Q3 – Q1). The IQR is important because it measures the spread of the middle 50% of the data, providing a robust measure of variability that is less affected by outliers compared to the range.
What is the formula for calculating the area of a triangle?
What is the formula for calculating the area of a triangle?The area of a triangle can be calculated using the formula: Area = 0.5 * base * height. Alternatively, for a triangle with sides of lengths a, b, and c, and semi-perimeter s = (a + b + c) / 2, Heron’s formula can be used: Area = sqrt(s * (s – a) * (s – b) * (s – c)).
How do you solve the equation $2x^2 + 3x – 5 = 0 $using the quadratic formula?
How do you solve the equation 2x^2 + 3x – 5 = 0 using the quadratic formula?To solve the quadratic equation 2x^2 + 3x – 5 = 0 using the quadratic formula, we use the formula x = [-b ± sqrt(b^2 – 4ac)] / 2a. Here, a = 2, b = 3, and c = -5. Plugging in these values, we get x = [-3 ± sqrt(3^2 – 4*2*(-5))] / (2*2). Simplifying, we find x = [-3 ± sqrt(9 + 40)] / 4, or x = [-3 ± sqrt(49)] / 4. Therefore, x = [-3 ± 7] / 4, giving us two solutions: x = 1 and x = -5/2.
What are the techniques for evaluating definite integrals using the Fundamental Theorem of Calculus?
What are the techniques for evaluating definite integrals using the Fundamental Theorem of Calculus?The Fundamental Theorem of Calculus connects differentiation and integration, providing a method to evaluate definite integrals. To use it, find the antiderivative F(x) of the integrand f(x). Then, evaluate F(b) – F(a), where [a, b] is the interval of integration. This result gives the net area under the curve f(x) from a to b.
What is the solution to the equation 3x + 5 = 20?
What is the solution to the equation 3x + 5 = 20?To solve the equation 3x + 5 = 20, first subtract 5 from both sides to get 3x = 15. Then, divide both sides by 3 to find x = 5. Thus, the solution is x = 5.
What are the basic steps for adding and subtracting fractions with different denominators?
What are the basic steps for adding and subtracting fractions with different denominators?To add or subtract fractions with different denominators, follow these steps: 1) Find the least common denominator (LCD) of the fractions. 2) Convert each fraction to an equivalent fraction with the LCD. 3) Add or subtract the numerators, keeping the denominator the same. 4) Simplify the resulting fraction if possible.
How do you solve a linear equation in one variable?
How do you solve a linear equation in one variable?To solve a linear equation in one variable, isolate the variable on one side of the equation using inverse operations. Simplify both sides of the equation as needed by combining like terms and performing arithmetic operations. The solution is the value of the variable that makes the equation true.
What are the basic steps to solve a linear equation?
What are the basic steps to solve a linear equation?To solve a linear equation, follow these steps: 1) Simplify both sides of the equation by combining like terms. 2) Move variable terms to one side and constant terms to the other using addition or subtraction. 3) Isolate the variable by dividing or multiplying both sides by the coefficient of the variable.
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Find the values of sin, cos, and tan for angles that satisfy the equation 2sin(x)cos(x) = 1
Answer 1 First, recognize that $2\sin(x)\cos(x) = \sin(2x)$. Thus, the equation becomes: $\sin(2x) = 1$ The solution for $\sin(2x) = 1$ occurs at: $2x = \frac{\pi}{2} + 2k\pi$, where $k$ is any integer. Thus: $x = \frac{\pi}{4} + k\pi$ For $k = 0$:...
Find the coordinates of a point on the negative unit circle given a specific angle
Answer 1 To find the coordinates of a point on the negative unit circle given a specific angle $ \theta $, we use the equation of the unit circle: $ x^2 + y^2 = 1 $ The coordinates can be found using parametric equations: $ x = -\cos(\theta) $ $ y =...
Determine the points on the negative unit circle where the tangent line is vertical
Answer 1 The negative unit circle is described by the equation:$ x^2 + y^2 = -1 $To find where the tangent line is vertical, we need to find the points where the derivative of $ y $ with respect to $ x $ is undefined. First, implicitly differentiate...
Prove that the sum of the squares of the sine and cosine functions on the unit circle equals 1
Answer 1 On the unit circle, any point is represented as $(\cos(\theta), \sin(\theta))$, where $\theta$ is the angle formed with the positive x-axis.According to the Pythagorean theorem, the equation of the unit circle is:$ x^2 + y^2 = 1...
Find the coordinates of the point on the unit circle for angle π/3
Answer 1 For the angle $ \frac{\pi}{3} $ on the unit circle, the coordinates are found using the sine and cosine functions.The x-coordinate is:$ \cos\left( \frac{\pi}{3} \right) = \frac{1}{2} $The y-coordinate is:$ \sin\left( \frac{\pi}{3} \right) =...
Determine the value of cos(7π/6) using the unit circle
Answer 1 To determine the value of $ \cos\left(\frac{7\pi}{6}\right) $ using the unit circle, we need to locate the angle $ \frac{7\pi}{6} $ in radians. This angle is in the third quadrant.In the third quadrant, the cosine function is negative. The...