Find sinθ, cosθ, and tanθ of the following right triangle ABC: AB = 3, BC = 4, ∠B = 90°, θ = ∠A.
Always start by drawing a picture! Label the vertices A, B, and C. Use the Pythagorean Theorem to the hypotenuse length (we will need this to find sin and cos). Let side AB be a, BC be b, and AC be c. a^2 + b^2 = c^2 3^2 + 4^2 = c^2 (Plug in a = 3 and b = 4) 9 + 16 = c^2 (Square) 25 = c^2 (Add) 5 = c (Square root) Now find sinθ, cosθ, and tanθ: sinθ = opposite/hypotenuse = 4/5 cosθ = adjacent/hypotenuse = 3/5 tanθ = opposite/adjacent = 4/3
The radius of a circle is 4 feet. What is the area of a sector bounded by a 135° arc?
The formula for the area of a sector is K = (m/360) * A, where K is the area of the sector, A is the area of the circle, and m is the measure in degrees of the arc bounding the sector. The sector's area depends on the arc's measure and the circle's area. You already know that the arc's measure is 135°, so find the circle's area. A = 𝜋r^2 = 𝜋(4)^2 (Plug in r = 4) = 16𝜋 (Square) So the area of the circle is 16𝜋 square feet. Now, find the area of the sector. K = (m/360) * A K = (135/360) * 16𝜋 (Plug in m = 135 and A = 16𝜋) K = 6𝜋 (Simplify) The area of the sector is 6𝜋 square feet.
Write a system of equations to describe the situation below and solve using substitution. Tanner and Sadie are both selling cookie dough for a fundraiser. Although Tanner has already sold 10 tubs, Sadie hasn't sold any yet. If Tanner starts selling 5 tubs per day and Sadie begins selling 7 tubs per day, they will eventually sell the same amount of cookie dough. How many days will that take? How many tubs will each sell?
To write a system of equations and solve, follow these steps: 1. Define your variables. 2. Write your system of equations. 3. Isolate a variable. 4. Plug the result of Step 3 into the other equation and solve for one variable. 5. Plug the result of Step 4 into one of the original equations and solve for the other variable. 6. State the solution. Following these steps: 1. Let x = days, y = tubs. 2. y = 5x + 10, y = 7x 3. The variable y is already isolated so we can skip this step! 4. Plug y = 5x + 10 into the other equation, y = 7x, and find the value of x. y = 7x 5x + 10 = 7x (Plug in y = 5x + 10) 10 = 2x (Subtract 5x from both sides) 5 = x (Divide both sides by 2) 5. Plug x = 5 into the one of the original equations (let's use y = 5x + 10), and find the value of y. y = 5x + 10 = 5(5) + 10 (Plug in x = 5) = 25 + 10 (Multiply) = 35 (Add) 6. Since x = 5 and y = 35, we can state the solution as follows: in 5 days, each will sell 35 tubs of cookie dough.