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# Tutor profile: Ashwin A.

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Ashwin A.
Experienced Tutor with a Passion for the High School Math Curriculum and Accounting
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## Questions

### Subject:Trigonometry

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Question:

Determine the exact value in radians for x, given that sin (x + pi/4) = 1/2, with 0 <=x<=2pi.

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Ashwin A.

Using trigonometric identity sin (x+y) = sinxcosy + sinycosx sin (x + pi/4) = sin(x)cos(pi/4) + sin(pi/4)cos(x) pi/4 is special triangle; using ratio of 1,1, root 2, sin pi/4 = 1/root 2 and cos pi/4 = 1/root 2 sin(x)* (1/root 2) + cos(x)*(1/root 2) = 1/2 sin(x)/root 2 + cos(x)/root 2 = 1/2 (sin(x) + cos(x))/root 2 = 1/2 Square both sides: ((sin(x) + cos(x))/root 2)^2 = (1/2)^2 Expand using perfect square trinomial method - note: (x+y)^2 = x^2 + 2xy + y^2 Likewise, (sin(x) + cos(x))^2 = sin(x)^2 + 2sin(x)cos(x) + cos(x)^2 ((sinx)^2 + 2sinxcosx + (cosx)^2)/2 = 1/4 (sinx)^2 + 2sinxcosx + (cosx)^2 = 1/2 Apply Pythagorean identity: sinx^2 + cosx^2 = 1 1 + 2sinxcosx = 1/2 Substitute trigonometric identity 2sinxcosx = sin (2x) 1 + sin (2x) = 1/2 sin (2x) = -1/2 2x = sin -1 (-1/2) - use inverse sine function Using special triangle ratio of 1, root 3, 2: sin pi/3 = root 3/2, sin pi/6 = 1/2; Using CAST rule, negative sine occurs in Quad 3 and Quad 4; therefore, sin -1 (-1/2) = 7pi/6 (Quadrant 3), 11pi/6 (Quadrant 4) - (Assuming 0 <= x <= 2pi) 2x = -7pi/6, -11pi/6 x = -7pi/6 * 1/2 and x = -11pi/6 * 1/2 x = 7pi/12, 11pi/12 Proof: sub x = 7pi/12 into original equation: sin (7pi/12 + pi/4) = sin (7pi/12 + 3pi/12) = sin (10pi/12) = sin (5pi/6) = 1/2 sub x = 11pi/12 into original equation: sin (-11pi/12 + pi/4) = sin (11pi/12 + 3pi/12) = sin (14pi/12) = sin (7pi/6) = -1/2 Therefore, of the two x values 7pi/12 and 11pi/12, 7pi/12 is the only value that satisfies the equation sin (x + pi/4) = 1/2. Therefore, x = 7pi/12.

### Subject:Calculus

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Question:

A ladder is being pulled up at a rate of 0.2 m per second till it reaches the foot of the roof to allow the construction worker to repair the roof shingles. The ladder is 5m long, leaning against the wall as it is pulled up to align with the roof. If the ladder reaches the roof when the wall is 4m high, how fast is the bottom of the ladder moving towards the wall?

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Ashwin A.

### Subject:Algebra

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Question:

A rectangle has a perimeter of 64 centimetres. Its area is 135 square centimetres, while its length is 12 cm more than 3 times its width. Determine the length and width of the rectangle in question.

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Ashwin A.

Given: A = 135 cm ^2 P = 64 cm length = 3 times width + 12 Define Variables: Let "w" represent the width of the rectangle. The length expressed in terms of the width is thus 3*w + 12 Required: Find the length and width. Solve for w, and determine 3w+12 (length as determined above). Determine Function: Rectangle Area = length * width Substitute terms & variables: 135 cm ^2 = w * (3w + 12) 135 = 3w^2 + 12w Rectangle Perimeter = 2 (length + width) Substitute terms & variables: 64 cm = 2 * (w + 3w + 12) * Expand and simplify by FOIL of brackets 64 = 2 * (4w + 12) 64 = 8w + 24 Divide each term on both sides by common factor of 8 to simplify 8 = w + 3 Given two equations: Perimeter: w + 3 = 8 Area: 3w^2 + 12w = 135 Determine 'w': Set equations equal to zero: w - 5 = 0 3w^2 + 12w - 135 = 0 Set equations equal to one another: w - 5 = 3w^2 + 12w - 135 Re-arrange the combined equation to set it equal to zero to set up for factoring a quadratic trinomial. 3w^2 + 12w - w - 135 + 5 3w^2 + 11w - 130 = 0 Use complex trinomial factoring: 3w^2 = 3w * w Two factors of 130 that add/subtract to 11w = 26, -5*3 (-15) (w - 5) (3w + 26) = 0 w-5 = 0 and 3w + 26 = 0 w = 5, -26/3 -26/3<0, therefore the only option for the "w" (width) value is w = 5 cm. If w = 5, substitute w = 5 into length = 3w + 12 3*(5) + 12 = 15 + 12 = 27 cm Therefore, given the above parameters in question, a rectangle of dimensions of 27cm x 5 cm would satisfy the above conditions.

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