Sum and difference formulae for sine and cosine Consider angles and - - PowerPoint PPT Presentation

Elementary Functions

Part 5, Trigonometry Lecture 5.1a, Sum and Difference Formulas

• Dr. Ken W. Smith

Sam Houston State University

2013

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Sum and difference formulae for sine and cosine

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).

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Sum and difference formulae for sine and cosine

The distance between P and Q is, by the Pythagorean theorem, the square root of d(PQ)2 = (cos α − cos β)2 + (sin α − sin β)2.

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Sum and difference formulae for sine and cosine

We can expand this out, algebraically, and get d(PQ)2 = (cos2 α−2 cos α cos β+cos2 β)+(sin2 α−2 sin α sin β+sin2 β). which we can rewrite (using the Pythagorean identity) as d(PQ)2 = 2 − 2 cos α cos β − 2 sin α sin β

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Sum and difference formulae for sine and cosine

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.

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Sum and difference formulae for sine and cosine

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Sum and difference formulae for sine and cosine

The distance from P ′ to Q′ is the square root of d(P ′Q′)2 = (1−cos(α−β))2+(sin(α−β))2 = 1−2 cos(α−β)+cos2(α−β)+sin2(α−β)) = 2 − 2 cos(α − β).

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Sum and difference formulae for sine and cosine

Since the line segments PQ and P ′Q′ are congruent, then we know that d(PQ)2 = d(P ′Q′)2.

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Sum and difference formulae for sine and cosine

In mathematics, if we arrive at the same value through two different computations, we always have valuable information. Here we can equate d(PQ)2 and d(P ′Q′)2 and simplify.

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Sum and difference formulae for sine and cosine

D2 = 2 − 2 cos α cos β − 2 sin α sin β D2 = 2 − 2 cos(α − β)2

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Sum and difference formulae for sine and cosine

So 2 − 2 cos α cos β − 2 sin α sin β = 2 − 2 cos(α − β)2. We may divide both sides by 2 and solve for cos(α − β). cos(α − β) = cos α cos β + sin α sin β This is an important result.

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Sum and difference formulae for sine and cosine

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)

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Sum and difference formulae for sine and cosine

If we note that sin(θ) = cos(π 2 − θ) (since sine and cosine are complementary functions!) then we can write sin(α + β) = cos(π 2 − (α + β)) = cos((π 2 − α) − β). Using the sum-of-angles formula above, with regards to the angles π 2 − α and β, we have sin(α + β) = cos((π 2 − α) − β) = cos(π 2 − α) cos β + sin(π 2 − α) sin β = sin α cos β + cos α sin β.

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Sum and difference formulae for sine and cosine

Therefore we have a sum-of-angles equation for the sine function: sin(α + β) = sin α cos β + cos α sin β (2) 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

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Sum and difference formulae for sine and cosine

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.

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Sum and difference formulae for sine and cosine

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

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Sum and difference formulae for sine and cosine

Here they are as difference of angles cos(α−β) = cos α cos β+ sin α sin β sin(α−β) = sin α cos β− cos α sin β

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Sum and Difference Formulas

In the next presentation, we will look at some applications of these sum and difference formulas. (End)

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Elementary Functions

Part 5, Trigonometry Lecture 5.1b, Applications of the Sum and Difference Formulas

• Dr. Ken W. Smith

Sam Houston State University

2013

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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 √ a2+b2 , sin θ = b √ a2+b2 .

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A reduction formula

By our sum-of-angles formulas, sin(x + θ) = sin x cos θ + cos x sin θ. If we replace cos θ and sin θ by

a √ a2+b2 and sin θ = b √ a2+b2 , we get

sin(x + θ) = sin x

a √ a2+b2 + cos x b √ a2+b2 = a sin x+b cos x √ a2+b2

. We clear the denominators by multiplying all sides by √ a2 + b2. a sin x + b cos x = √ a2 + b2 sin(x + θ). (3)

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Sum and difference formulae for sine and cosine

a sin x + b cos x = √ a2 + b2 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 =

• 12 +

√ 3

2 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

• 12 +

√ 3

2 = √1 + 3 = 2, we have

sin x + √ 3 cos x = 2 sin(x + π/3).

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Sum and difference formulas 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)

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Sum and difference formulae for tangent

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

• btain

tan(α − β) = tan α − tan β 1 + tan α tan β .

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Some worked problems

Compute the exact value of cos 75◦. Solution. Using the sum of angles formula for cosine and the fact that 75◦ = 45◦ + 30◦, we have cos 75◦ = cos(45◦ + 30◦) = cos 45◦ cos 30◦ − sin 45◦ sin 30◦ = √ 2 2 √ 3 2 − √ 2 2 1 2 = √ 6 − √ 2 4

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Sum and difference formulae for sine and cosine

Compute the exact value of sin 15◦. Solution. sin 15◦ = sin(45◦ − 30◦) = sin 45◦ cos 30◦ − cos 45◦ sin 30◦ = √ 2 2 √ 3 2 − √ 2 2 1 2 = √ 6 − √ 2 4 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!)

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Sum and difference formulae for sine and cosine

Find the tangent of 75◦.

• Solution. We could separately compute the sine and cosine of 75◦ using
• ur sum-of-angle formulas (as we did on the last slides) or we could use

the sum-of-angles formula for tangent and write tan 75◦ = tan(45◦ + 30◦) = tan 45◦ + tan 30◦ 1 − tan 30◦ tan 45◦ = 1 + √ 3 3 1 − ( √ 3 3 )(1) . Multiplying numerator and denominator by 3 gives tan 75◦ = 3 + √ 3 3 − √ 3 . If we don’t like the square roots in the denominator, we can multiply numerator and denominator by 3 + √ 3 (the conjugate of the denominator) and find that tan 75◦ = (3 + √ 3 3 − √ 3)(3 + √ 3 3 + √ 3) = (9 + 6 √ 3 + 3 9 − 3 ) = (12 + 6 √ 3 6 ) = 2 + √ 3 .

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Sum and Difference Formulas

In the next presentation, we will look at double angle and power reduction formulas. (End)

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