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Unformatted text preview: MATH 137 WEEK 2 NOTES PROF. DOUG PARK radians Measuring a = 1 is the equation of the “unit circle” of radius 1 cen umference of unit circle is 2 π . There is a one s P on x 2 + y 2 = 1 } x y P θ 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 xaxis, the angles are given a minus sign. 1 x y π/3π/3 = 5π/3 ic functions A x y P θ Def . P = (cos θ, sin θ ) i.e. cos θ = [ xcoordinate of P ] sin θ = [ ycoordinate of P ] [Domain of cos θ and sin θ ] = R [Range of cos θ and sin θ ] = [ 1 , 1] Date : September 25, 2009. 1 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 MATH 137 3 Terminology : cos θ and sin θ have lower bound 1 and upper bound 1. Periodicity. Let Z = { , ± 1 , ± 2 ,... } = { integers } . For any k ∈ Z , and any fixed trig function, trig( θ 2 kπ ) = 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 kπ 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 2  θ θ 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 4 DOUG PARK 5. Graphs of trig functions Graph of sin x ....
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This note was uploaded on 12/18/2010 for the course ECONOMICS 120 taught by Professor Mesta during the Spring '10 term at Wilfred Laurier University .
 Spring '10
 Mesta

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