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1007_Tut1_Sol - MATH 1007 A Tutorial#1 SOLUTIONS Example of...

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MATH 1007 A Tutorial #1 Sept 22, 2006 SOLUTIONS Example of a proof: To show that 2 csc 2 x = sec x csc x , we start with one side, say the left side (LS), and show that it is equal to the other side: LS = 2 csc 2 x = 2 sin 2 x = 2 2 sin x cos x = 1 sin x cos x = 1 sin x 1 cos x = csc x sec x = RS 1. [2 marks] Prove the following trigonometric identity, using the approach above. (sin x + cos x ) 2 = 1 + sin 2 x Solution: LS = (sin x + cos x ) 2 = sin 2 x + 2 sin x cos x + cos 2 x = (sin 2 x + cos 2 x ) + 2 sin x cos x = 1 + 2 sin x cos x = RS 1
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2. [2 marks] Find the slope and the equation of the tangent line to the curve y = x 3 at the point where x = - 1. Solution: y = 3 x 2 y ( - 1) = 3( - 1) 2 = 3 Since ( - 1) 3 = - 1, the point ( - 1 , - 1) is on y = x 3 . We now find the equation of the line through ( - 1 , - 1) with slope 3: y - y 1 = m ( x - x 1 ) y - ( - 1) = 3( x - ( - 1)) y + 1 = 3 x + 3 y = 3 x + 2 Therefore the equation of the tangent line to y = x 3 at the point where x = - 1 is y = 3 x + 2, and the slope of this tangent line is 3. 3. [4 marks] If sin x = 12 13 and sec y = 5 4 , where x & y lie between 0 and π 2 , calculute sin( x + y ). Hint: Use the given information to build 2 right triangles, one with angle x and one with angle y . You know 2 of the 3 sides of the triangle by using the SOH-CAH-TOA rule. Now use the pythagorean theorem to solve for the 3rd side of the triangle.
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