Math

How do you find the values of the sine, cosine, and tangent functions for common angles like 30°, 45°, and 60°?To find the sine, cosine, and tangent values for 30°, 45°, and 60°, use the special right triangles: 30°-60°-90° and 45°-45°-90°. For 30°: sin=1/2, cos=√3/2, tan=1/√3. For 45°: sin=cos=√2/2, tan=1. For 60°: sin=√3/2, cos=1/2, tan=√3.

In a study, researchers collected data on the participants’ age, income, and the number of hours they exercise each week. How can we use multiple regression analysis to assess the combined effect of age and income on the time spent exercising, including eTo assess the combined effect of age and income on exercise time using multiple regression analysis, we can model exercise hours as the dependent variable and age and income as independent variables. First, verify assumptions: linearity, homoscedasticity, normality of residuals, and independence. Check for multicollinearity using Variance Inflation Factor (VIF); VIF > 10 indicates significant multicollinearity. Address issues by removing or combining variables if necessary. Evaluate model fit using R-squared and adjusted R-squared.

How do you solve systems of equations using the substitution method?To solve systems of equations using the substitution method, first solve one equation for one variable. Substitute this expression into the other equation. Solve for the remaining variable, then back-substitute to find the other variable. Verify the solution by plugging the values into the original equations.

What is the property of the operations used in solving the equation: 2x + 5 = 15?The properties used in solving the equation 2x + 5 = 15 include the properties of equality, specifically the subtraction property of equality (subtracting 5 from both sides) and the division property of equality (dividing both sides by 2). These operations maintain the balance of the equation.

How do you solve a system of equations using the substitution method?To solve a system of equations using the substitution method, first solve one equation for one variable in terms of the other variable. Then substitute this expression into the other equation. Solve for the remaining variable and substitute back to find the first variable. Verify the solution by plugging the values into the original equations.

How do you determine the end behavior of a polynomial function?To determine the end behavior of a polynomial function, examine the leading term, which is the term with the highest degree. The sign and degree of this term dictate the end behavior. For example, if the leading term is positive and has an even degree, both ends of the graph rise. If it is negative and even, both ends fall. For odd degrees, if positive, the left end falls and the right end rises; if negative, the left end rises and the right end falls.

What is the limit of a function as x approaches a particular value, and how do you find it?The limit of a function as x approaches a particular value is the value that the function approaches as the input gets arbitrarily close to that point. To find it, evaluate the function as x approaches the value from both sides. If the values converge to the same number, that is the limit.

How can you use the unit circle to find the trigonometric values for any angle?The unit circle, with a radius of 1 centered at the origin of the coordinate plane, is a powerful tool for finding trigonometric values of any angle. By defining an angle θ in standard position, where its vertex is at the origin and its initial side lies along the positive x-axis, the terminal side of the angle intersects the unit circle at a specific point (x, y). The x-coordinate of this point represents cos(θ), while the y-coordinate represents sin(θ). For tangent, tan(θ) is found by dividing the sine by the cosine (tan(θ) = sin(θ)/cos(θ)). This method can be extended to angles beyond 0° to 360° by considering their coterminal angles or using symmetry properties of the unit circle.

How do you solve equations that involve both fractions and decimals multiple terms on both sides of the equation?To solve such equations, first convert all fractions and decimals to a common form, usually fractions. Then, find a common denominator for all fractions. Clear the fractions by multiplying through by the least common denominator. Simplify the resulting equation and solve for the variable. Finally, check your solution.

What is the difference between mean and median, and when should each be used?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. Use the mean for symmetric distributions without outliers; use the median for skewed distributions or when outliers are present.