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Week 2 Course Notes - MATH 137 WEEK 2 NOTES PROF DOUG PARK...

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MATH 137 WEEK 2 NOTES PROF. DOUG PARK 1. Angles in radians Measuring angles in radians. x 2 + y 2 = 1 is the equation of the “unit circle” of radius 1 centered at the origin. The circumference of unit circle is 2 π . There is a one-to-one correspondence: { Angles θ } ←→ { Points P on x 2 + y 2 = 1 } 1 A x y P θ O O = (0 , 0) A = (1 , 0) θ = ] AOP (measured ) = length of _ AP Def . θ = radians Ex . 360 = 2 π , 180 = π , 90 = π 2 , 60 = π 3 , 45 = π 4 , 30 = π 6 , 0 = 0 Negative angles. When measured in clockwise direction from the positive x -axis, the angles are given a minus sign. 1 x y π/3 -π/3 = 5π/3 2. Trigonometric functions 1 A x y P θ O Def . P = (cos θ, sin θ ) i.e. cos θ = [ x -coordinate of P ] sin θ = [ y -coordinate of P ] [Domain of cos θ and sin θ ] = R [Range of cos θ and sin θ ] = [ - 1 , 1] Date : September 25, 2009. 1
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2 DOUG PARK Def . tan θ = sin θ cos θ = [slope of OP ] , sec θ = 1 cos θ , csc θ = 1 sin θ , cot θ = 1 tan θ Definition via right triangle. When 0 < θ < π 2 , the previous definitions agree with the right triangle definition of cos θ and sin θ . 1 x y P = ( a , b ) θ O a b cos θ = base hypotenuse = a 1 = a, sin θ = height hypotenuse = b 1 = b We have thus generalized the triangle definition of cos θ and sin θ to angles that are not between 0 and π 2 . Triangles to remember. 1 1 1 π/4 π/4 π/4 π/4 2 1 1 2 2 1 π/6 π/3 π/3 π/6 2 1 2 1 2 3 3 cos π 4 = sin π 4 = 1 2 , tan π 4 = 1 cos π 6 = 3 2 = sin π 3 , sin π 6 = 1 2 = cos π 3 tan π 6 = 1 3 , tan π 3 = 3 3. Properties of trig functions Since (cos θ, sin θ ) lies on the unit circle x 2 + y 2 = 1, cos 2 θ + sin 2 θ = 1 We usually write (cos θ ) 2 = cos 2 θ , etc. Since the coordinates of points on the unit circle lie between - 1 and 1, - 1 cos θ 1 , - 1 sin θ 1
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MATH 137 3 Terminology : cos θ and sin θ have lower bound - 1 and upper bound 1. Periodicity. Let Z = { 0 , ± 1 , ± 2 , . . . } = { integers } . For any k Z , and any fixed trig function, trig( θ - 2 ) = trig θ where trig is any one of 6 trig functions: cos , sin , tan , sec , csc , cot This is because angles θ and θ - 2 correspond to the same point on the unit circle! Even/odd. x y θ - θ (cos θ , sin θ ) (cos(- θ) , sin(- θ) ) From the picture, we deduce that cos( - θ ) = cos θ = cos x is even function. sin( - θ ) = - sin θ = sin x is odd function. 4. sin and cos are essentially the same π/2 - θ θ b a c cos θ = a c = sin π 2 - θ sin θ = b c = cos π 2 - θ Analytic Continuation Theorem . Any trigonometric identity that holds for the interval 0 < θ < π 2 remains true for all angle θ R for which the functions are defined. Thus for all x R , cos x = sin π 2 - x = - sin x - π 2 sin x = cos π 2 - x = cos x - π 2
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4 DOUG PARK 5. Graphs of trig functions Graph of sin x .
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