Sum and difference formulae for sine and cosine. Elementary Functions. Consider angles α and β with α > β. These angles identify points on the

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1 Consider angles α and β with α > β. These angles identify points on the unit circle, P (cos α, sin α) and Q(cos β, sin β). Part 5, Trigonometry Lecture 5.1a, Sum and Difference Formulas Dr. Ken W. Smith Sam Houston State University 1 / 8 The distance between P and Q is, by the Pythagorean theorem, the square root of We can expand this out, algebraically, and get d(p Q) = (cos α cos β) + (sin α / 8 d(p Q) = (cos α cos α cos β +cos β)+(sin α sin α sin β +sin β). sin β). which we can rewrite (using the Pythagorean identity) as d(p Q) = cos α cos β sin α sin β 3 / 8 4 / 8

2 Let s rotate this picture clockwise through the angle β so that the point Q becomes Q (1, 0) lying on the x-axis. The point P rotates into the point P (cos(α β), sin(α β)) The distance from P to Q is the same as the distance from P to Q. 5 / 8 The distance from P to Q is the square root of d(p Q ) = (1 cos(α β)) +(sin(α β)) = 1 cos(α β)+cos (α β)+sin (α β)) = cos(α β). 6 / 8 Since the line segments P Q and P Q are congruent, then we know that d(p Q) = d(p Q ). 7 / 8 8 / 8

3 D = cos α cos β sin α sin β In mathematics, if we arrive at the same value through two different computations, we always have valuable information. Here we can equate d(p Q) and d(p Q ) and simplify. 9 / 8 10 / 8 D = cos(α β) So cos α cos β sin α sin β = cos(α β). We may divide both sides by and solve for cos(α β). cos(α β) = cos α cos β + sin α sin β This is an important result. We create a more memorable form of this equation if we replace β by β and use the fact that sine is an odd function while cosine is an even function. Write cos(α + β) = cos(α ( β)) and replace β by β in our equation for cos(α β) to see that cos(α + β) = cos α cos( β) + sin α sin( β) = cos α cos(β) sin α sin(β) This is an equation we want to record and use on a regular basis. cos(α + β) = cos α cos β sin α sin β (1) 11 / 8 1 / 8

4 If we note that sin(θ) = cos( π θ) (since sine and cosine are complementary functions!) then we can write sin(α + β) = cos( π (α + β)) = cos((π α) β). Using the sum-of-angles formula above, with regards to the angles π α and β, we have sin(α + β) = cos(( π α) β) Therefore we have a sum-of-angles equation for the sine function: sin(α + β) = sin α cos β + cos α sin β () If we need, we may replace β by β to create a difference equation: sin(α β) = sin α cos β cos α sin β = cos( π α) cos β + sin(π α) sin β = sin α cos β + cos α sin β. 13 / 8 14 / 8 I don t memorize these identities. (That s one reason I ve been attracted to mathematics if one understands the math, one doesn t need to memorize!) In the undergraduate classes that I teach at Sam Houston State University, I will provide students with the sum of angle formulas when they are needed. Here they are again: cos(α + β) = cos α cos β sin α sin β sin(α + β) = sin α cos β + cos α sin β 15 / 8 16 / 8

5 Sum and Difference Formulas Here they are as difference of angles In the next presentation, we will look at some applications of these sum and difference formulas. cos(α β) = cos α cos β+ sin α sin β (End) sin(α β) = sin α cos β cos α sin β 17 / 8 18 / 8 0 / 8 A reduction formula Let P (cos θ, sin θ) be a point on the terminal side of angle θ. Any point (a, b) on the line OP satisfies the equations cos θ = a, a +b sin θ = b. a +b Part 5, Trigonometry Lecture 5.1b, Applications of the Sum and Difference Formulas Dr. Ken W. Smith Sam Houston State University 19 / 8

6 A reduction formula By our sum-of-angles formulas, sin(x + θ) = sin x cos θ + cos x sin θ. a If we replace cos θ and sin θ by and sin θ = b a +b a, we get +b a sin(x + θ) = sin x + cos x b a sin a = x+b cos x +b a. +b a +b We clear the denominators by multiplying all sides by a + b. a sin x + b cos x = a + b sin(x + θ). (3) a sin x + b cos x = a + b sin(x + θ). This formula is useful for changing a linear combination of sine and cosine functions into just a sine function. For example, the function f(x) = sin x + 3 cos x can be rewritten as sin x + 3 cos x = sin(x + θ) where θ is the angle between the x-axis and the line from the origin to the point (1, 3). Since θ = π/3 in this problem, and since = =, we have sin x + 3 cos x = sin(x + π/3). 1 / 8 Sum and difference formulas for tangent / 8 Sum and difference formulae for tangent We know that tan(α + β) = sin(α+β) cos(α+β) so we may use our sum-of-angle formulas to create a formula for tan(α + β). A first pass gives tan(α + β) = sin(α+β) cos(α+β) = sin α cos β+cos α sin β cos α cos β sin α sin β But we would really like a sum-of-angles formula for tangent that is in terms of tan α and tan β. So let s divide both numerator and denominator by cos α cos β. tan(α + β) = tan α + tan β 1 tan α tan β. (4) tan(α + β) = tan α + tan β 1 tan α tan β. What if we wanted an equation for tan(α β)? Replace β by β and use the fact that tangent is an odd function to obtain tan(α β) = tan α tan β 1 + tan α tan β. 3 / 8 4 / 8

7 Some worked problems Compute the exact value of cos 75. Solution. Using the sum of angles formula for cosine and the fact that 75 = , we have cos 75 = cos( ) = cos 45 cos 30 sin 45 sin 30 = 3 1 = 6 4 Compute the exact value of sin 15. Solution. sin 15 = sin(45 30 ) = sin 45 cos 30 cos 45 sin 30 = 3 1 = 6 This answer looks familiar! (Notice that since 75 and 15 are complementary angles so the cosine of one angle is the sine of the other!) 4 5 / 8 6 / 8 Sum and Difference Formulas Find the tangent of 75. Solution. We could separately compute the sine and cosine of 75 using our sum-of-angle formulas (as we did on the last slides) or we could use the sum-of-angles formula for tangent and write 3 tan 75 = tan( ) = tan 45 + tan tan 30 tan 45 = ( 3 )(1) Multiplying numerator and denominator by 3 gives tan 75 = If we don t like the square roots in the denominator, we can multiply numerator and denominator by (the conjugate of the denominator) and find that tan 75 = ( )( ) = ( ) = ( ) = + 7 / In the next presentation, we will look at double angle and power reduction formulas. (End) 8 / 8

Trig Identities. or (x + y)2 = x2 + 2xy + y 2. Dr. Ken W. Smith Other examples of identities are: (x + 3)2 = x2 + 6x + 9 and

Trig Identities. or (x + y)2 = x2 + 2xy + y 2. Dr. Ken W. Smith Other examples of identities are: (x + 3)2 = x2 + 6x + 9 and Trig Identities An identity is an equation that is true for all values of the variables. Examples of identities might be obvious results like Part 4, Trigonometry Lecture 4.8a, Trig Identities and Equations