Tutor profile: Snezana T.
Prove trigonometric identity: cos α + cos 2α + cos 6α + cos 7α = 4 cos (α/2) cos (5α/2) cos (4α).
When proving trigonometric identities transform one side of the equation until equals the other. We will start from the left to the right. Mathematics tool that we will use is associative property, sum to product identity, and factoring. Sum to product factoring formula is: cos (A+B) = 2 cos [(A+B)/2] cos [(A-B)/2]. We will transform the left side using associative property of addition: cos α + cos 2α + cos 6α + cos 7α = (cos 7α + cos α) + (cos 6α + cos 2α) = Then we will apply sum to product identity formula: = 2 cos [(7α + α)/2]cos[(7α - α)/2] + 2 cos[( 6α + 2α)/2] cos[(6α -2α)/2]) = Simplify: = 2 cos (4α) cos (3α) + 2 cos (4α) cos (2α) = and factor = 2 cos (4α) (cos (3α) + cos (2α)) = Apply sum to product formula again: = 2 cos (4α) 2 cos[( 3α + 2α)/2] cos[(3α -2α)/2] = = 4 cos (α/2) cos (5α/2) cos (4α). Therefore: cos α + cos 2α + cos 6α + cos 7α = 4 cos (α/2) cos (5α/2) cos (4α).
The perimeter of a square is 7x – 3 inches. If the area of the square is 64 square inches, what is the value of x?
Let a be the side of the square. Let A be the area of the square. Let P be the perimeter of the square. Since A = 64 square inches, and A = a^2, the side of the square is: a = √A = √64 = 8 inches. Since the formula for the perimeter of a square is P = 4a and the perimeter is 7x -3 : 4a = 7x - 3. Therefore: 4٠8 = 7x – 3. By adding 3 to both sides we get: 35 = 7x. Therefore, x = 5.
A boat can travel 8 miles up a river in 2 hours. The same boat can travel 6 miles downstream in 30 minutes. What is the speed of the boat in still water? What is the speed of the current?
When solving word problems ask these questions: • What are we trying to find? • Which data are we given? • Which mathematical tools can we use to solve the problem? What are we trying to find? We want to find the speed of the boat in still water and the speed of the current. Let b be the speed of the boat in still water. Let c be the speed of the current. Which data are we given? Let’s organize the given data in the chart: Distance Time upstream 8 miles 2 hours downstream 6 miles ½ hour = 30 minutes Which mathematical tools can I use to solve the problem? We need to find rates of the boat and the current so we will use the formula: Distance = Rate X Time. We have two unknown variables, so we need two equations to solve the system. When going upstream, the stream direction is opposite to the direction of the boat, which means the stream is slowing the boat down. Therefore, the rate is b-c, so our first equation is: 8 = (b-c)(2) When going downstream, the stream direction is the same as the direction of the boat, so the stream is speeding the boat up. Therefore, the rate is b+c, so our second equation is: 6 = (b+c)(1/2) Now we solve the system of equations: 8 = (b-c)(2) 6 = (b+c)(1/2) We divide the first equation by 2, and multiply the second equation by 2 we get: 4 = b-c 12 = b+c We add these equations to find b: 16 = 2b b = 8 The speed of the boat in the still water is 8 miles per hour. We substitute b in any of the equations, and get: 12 = 8+c c = 4 The speed of the current is 4 miles per hour.
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