Prove that [ cos(x) - sin(x) ] [ cos(2x) - sin(2x) ] = cos(x) - sin(3x)
We will start with right hand side: cos(x) - sin(3x) = cos(x) - sin( x + 2x) // break down the sin (3x) so that we can apply standard formula Apply the formula for sin(A+B) = sinAcosB+cosBsinA = cos(x) - sin(x)cos(2x) - cos(x)sin(2x) We will now expand the left hand side: [ cos(x) - sin(x) ][ cos(2x) - sin(2x) ] = cos(x) cos(2x) - cos(x) sin(2x) - sin(x) cos(2x) + sin(x) sin(2x) Use the identities cos(2x) = 1 - 2 sin2 and sin(2x) = 2 sin(x) cos(x) to transform the first two terms (only) in the above expression. = cos(x)(1 - 2 sin2) + sin(x) 2 sin(x) cos(x) - cos(x) sin(2x) - sin(x) cos(2x) = cos(x) - 2 cos(x) sin2 + 2 cos(x) sin2 - cos(x) sin(2x) - sin(x) cos(2x) = cos(x) - cos(x) sin(2x) - sin(x) cos(2x) The left hand side has been transformed so that it is equal to the right hand side.
Calculate the area of a right angled isosceles triangle which has a hypotenuse equal to 24.
Since the two legs of a right isosceles triangle have equal lengths; let x be one of these lengths. The area A of the triangle is given by Formula used: Area = 0.5*base*height A = (1/2) x * x = (1/2)x2 Pythagoras' theorem: hypotenuse*hypotenuse = base*base + height*height We now use Pythagoras' theorem to find x as follows x2 + x2 = 242 Simplify 2 x2 = 576 x2 = 288 We now calculate the area A as follows A = (1/2)x2 = (1/2) 288 = 144
It takes 6 hours for Peter to travel from Los Angeles to San Jose. On his way back, if he increases his speed by 20 miles/hour, he takes 0.5 hours less than what he took. The distance between both the cities is 340 miles. Find the average speed for the whole journey.
Let x and x + 20 be the speeds of the car from Los Angeles to San Jose and then from San Jose to Los Angeles. Hence the distance from Los Angeles to San Jose may expressed as 6x and the distance from San Jose to Los Angeles as 5.5(x + 20) The average speed = total distance / total time = (6x + 5.5 (x + 20)) / (6 + 5.5) The distance from A to B is equal to the distance from B to A, hence: 6 x = 5.5(x + 20). Solve for x to obtain x = 220 miles/hr. We now substitute x by 220 in the formula for the average speed to obtain. average speed = 229.565 miles/hr Note: Here the information about the distance between the two places was redundant and was provided just to force the person to thoroughly think about what concept to apply and what not.