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Amit R.

Student at Columbia University

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SAT II Mathematics Level 1

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Question:

What is the value of the angle pie over 6 if the hypotenuse is 1.

Amit R.

Answer:

Remember, pie is equal to 180 degrees. So? we can take 180 and divide it by 6 to get the value of the angle. 180 divide by 6 = 30. In trig triangles, one angle has to be 90 (right-angled triangle). And since we discovered that the value of the other angle is 30, we know that 180-90-30= 60. That is because we have to remember that the sum of all the angles in a triangle is 180. This is also a triangle with an identity we need to recognize. In this triangle, the hypotenuse is 1 ( given in the question) and the opposite side to the 30 degrees angle is half of the hypotenuse, which in this case is 1/2. So? if we take the sin of 30 degrees, we get 1/2 over 1. which is 1/2. If you notice this is a great way to find out the actual value of sin(30), or sin(60) if you have a hard time to memorize their value. ( just make sure the value of the hypotenuse as 1).

Calculus

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Question:

what is the value of the integral of -3 to 3 of t^2*sinx

Amit R.

Answer:

At first glance, this question looks complicated. But, since sin(x) is an odd function. The integral of this function would be even ( in this case cos(x) ). That is a rule in Calculus. Just to give you an example of a simpler question take for instance the value of 3x^3 [ an odd function since it is in the power of 3]. Notwithstanding, its integral is x^2 which is even. For the question above, since the value of the original function is even(cox(x)). We need to remember the rule that in even functions f(-x) = f(x) . Which means that in this case f(-3) = f(3). Thus, to solve this question, we continue with part 1 of "Fundamental Theorem of Calculus " which means that the value of the original function is the value of 3 in the function minus the values of -3 in the function. So? cos(3) - cos(-3) = 0! ( because the value of 3 and -3 is similar, it is basically like doing 3 - 3 = 0).

Algebra

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Question:

The value of X is one more than the value of y. The value of 10,000Y is 1 more than Z. If x is 1, what is the Value of (3Z+1)?

Amit R.

Answer:

First, we want to understand the value of X since it will help us to get Y which would help us to get Z. ( X --> Y ---> Z: in order for us to realize it we have to look carefully at the question and write down all of the equations we have. That is the way to break down the question and build a full picture that will help us to find the solution). X = Y + 1 10,000Y = Z +1 3z+1? X= Y+1 ( since X is one more than Y, we want to add 1 to Y to make them EVEN). So? if x is 1, Y is : 1 = Y + 1 / -1 (moving 1 to the other side and flip the sign) Y = 1-1 Y= 0 Let's continue, so if 10,000Y is 1 more than Z, we write the equality 10,000Y = z + 1 ( Z gets one because he is the smaller one). Note: do not be afraid of 10,000. As we saw, Y is actually 0! Thus, even it would be 100,000,000, it does not matter! [Every integer times zero is zero]. so? 10,000 * 0 = Z +1 0 = Z +1 ( now let's move 1 to the other side, do not forget to flip its sign!) Z = -1 Now, let's check what the question wanted. 3z - 1. ( We found that z is -1). So? 3 * (-1) -1 = -3 -1 = -4 ( Make sure you multiple 3 and -1 correctly) The final answer is -4.

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