Math

How can I determine the exact values for the sine, cosine, and tangent of a 45-degree angle?To determine the exact values for sine, cosine, and tangent of a 45-degree angle, consider a right triangle with equal legs. The hypotenuse is √2 times the leg length. Thus, sin(45°) = cos(45°) = 1/√2 or √2/2, and tan(45°) = 1.

How do you prove that the sum of the angles of any triangle always equals 180 degrees using trigonometric functions and identities?To prove the sum of the angles of any triangle equals 180 degrees using trigonometric functions and identities, consider a triangle with angles A, B, and C. Using the identity for the tangent of the sum of two angles, tan(A + B) = (tan A + tan B) / (1 – tan A tan B). Since tan(C) = tan(180° – (A + B)) and tan(180° – x) = -tan(x), it follows that tan(A + B) = -tan(C). This implies that A + B + C = 180°.

How do you solve trigonometric equations that involve multiple angles, such as 2sin(x)cos(x) = sin(x), within the interval [0, 2π]?To solve 2sin(x)cos(x) = sin(x) within [0, 2π], first use the identity 2sin(x)cos(x) = sin(2x). The equation becomes sin(2x) = sin(x). This implies two cases: 2x = x + 2kπ or 2x = π – x + 2kπ. Solving these gives x = 0, π, 2π, π/3, 5π/3.

How do you find the amplitude and period of the function y = 3sin(2x – π)?To find the amplitude and period of the function y = 3sin(2x – π), first identify the amplitude, which is the coefficient of the sine function, here it is 3. The period is determined by the coefficient of x inside the sine function. The period of sin(Bx) is given by (2π)/B. For y = 3sin(2x – π), B is 2, so the period is (2π)/2 = π.

If a packet of stickers costs $1.25 and you have $18.75, how many packets of stickers can you buy, and how much money will you have left over?You can buy 15 packets of stickers with $18.75. Each packet costs $1.25. After purchasing 15 packets, you will have no money left over, as 15 packets x $1.25 per packet equals exactly $18.75.

How do I solve the equation 3(x + 2) equals 18 and what properties should I use to simplify the expression?To solve the equation 3(x + 2) = 18, first use the Distributive Property to expand it to 3x + 6 = 18. Then, apply the Subtraction Property of Equality to isolate the variable: 3x = 12. Finally, use the Division Property of Equality to solve for x: x = 4.

How do you find the vertex of a parabola given its equation in standard form?To find the vertex of a parabola given its equation in standard form y = ax^2 + bx + c, use the vertex formula: x = -b/(2a). Substitute this x-value back into the equation to find the corresponding y-value. The vertex is at the point (x, y).

How do I determine all the possible solutions for the equation sin(2x) = cos(x) within one period in radians?To find all solutions for sin(2x) = cos(x) within one period [0, 2π), use the identity sin(2x) = 2sin(x)cos(x). This converts the equation to 2sin(x)cos(x) = cos(x). Solving, we get cos(x)(2sin(x) – 1) = 0. The solutions are x = π/2, 3π/2, and x = π/6, 5π/6.

What is the difference between mean, median, and mode, and how do you calculate each?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, or the average of the two middle values if the set has an even number of values. The mode is the most frequently occurring value in the data set.

What is the difference between a sample and a population in statistics, and why is it important to use a sample when studying large groups?In statistics, a population is the entire group being studied, while a sample is a subset of that population. Using a sample is crucial when studying large groups because it is often impractical or impossible to collect data from every individual in the population. Samples allow for manageable, cost-effective, and timely analysis while still providing insights into the population as a whole.