Exercise 36 Page 524

We are told that the equation of a standing wave can be obtained by adding the displacements of two waves traveling in opposite directions. y= A cos (2π t/3-2π x/5 ) + A cos ( 2π t/3+2π x/5) Here, t represents the time and x is the point's position. We want to show that this formula can be rewritten as the following equation when t is equal to 1. y= - A cos (2π x/5 ) To do so, we will start by substituting t= 1 in the given formula.

y= A cos (2π t/3-2π x/5 ) + A cos ( 2π t/3+2π x/5)

Substitute

t= 1

y= A cos (2π ( 1)/3-2π x/5 ) + A cos ( 2π ( 1)/3+2π x/5)

IdPropMult

Identity Property of Multiplication

y= A cos (2π/3-2π x/5 ) + A cos ( 2π/3+2π x/5)

Let's recall the difference formula for cosine. cos ( a- b) = cos acos b + sin a sin b We can apply this to the first term of the obtained formula.

y=A cos (2π/3-2π x/5 ) + A cos ( 2π/3+2π x/5)

cos(α-β)=cos(α)cos(β)+sin(α)sin(β)

y=Acos ( 2π/3 ) cos ( 2π x/5 ) + Asin ( 2π/3 ) sin( 2π x/5 ) + A cos ( 2π/3+2π x/5)

This time let's recall the sum formula for cosine. cos ( a+ b) = cos acos b - sin a sin b We will use this formula to continue simplifying our equation.

y=Acos ( 2π/3 ) cos (2π x/5 ) + Asin (2π/3 ) sin(2π x/5 ) + A cos ( 2π/3+2π x/5)

cos(α+β)=cos(α)cos(β)-sin(α)sin(β)

y=Acos (2π/3) cos (2π x/5) + Asin (2π/3) sin (2π x/5) + Acos ( 2π/3)cos ( 2π x/5) - Asin ( 2π/3) sin ( 2π x/5)

AddSubTerms

Add and subtract terms

y=2Acos (2π/3) cos (2π x/5)

Now, we need to find the exact value of cos ( 2π3 ). To do so, let's graph θ = 2π3 in standard position so that we can find its reference angle. Note that the terminal side of this angle lies in Quadrant II. Therefore, to find its reference angle θ' we will subtract 2π3 from π.

An angle theta of 2pi/3 is drawn on a coordinate plane. On the other side of the angle, the supplementary angle is indicated by subtracting 2pi/3 from pi, which is equal to pi/3.

Next, we will recall the signs of the six trigonometric functions in the different quadrants of the coordinate plane.

The signs of every trigonometric function are indicated according to a coordinate plane. In Quadrant I, all trigonometric functions are positive. In Quadrant II, only the sine and the cosecant are positive. In Quadrant III, only the tangent and the cotangent are positive. In Quadrant IV, only the cosine and the secant are positive.

In Quadrant II — the quadrant where the terminal side of the angle is located — cosine is negative. With this information, we can write an equation relating the cosine of the angle and the cosine of its reference angle. cos (2π/3 )= - cos (π/3 ) Now we can recall some trigonometric values for special angles.

Trigonometric Values for Special Angles
θ	Sine	Cosine	Tangent	Cosecant	Secant	Cotangent
π/6	1/2	sqrt(3)/2	sqrt(3)/3	2	2sqrt(3)/3	sqrt(3)
π/4	sqrt(2)/2	sqrt(2)/2	1	sqrt(2)	sqrt(2)	1
π/3	sqrt(3)/2	1/2	sqrt(3)	2sqrt(3)/3	2	sqrt(3)/3

Using the table, we can see that cos π3= 12. cos (2π/3) = - cos (π/3) ⇓ cos (2π/3) = - 1/2 Finally, we will substitute this value into the obtained function to continue simplifying.

y=2Acos (2π/3) cos (2π x/5)

Substitute

cos (2π/3)= - 1/2

y=2A( - 1/2) cos (2π x/5)

▼

Simplify right-hand side

MoveLeftFacToNumOne

a* 1/b= a/b

y=- 2A/2 cos (2π x/5)

CrossCommonFac

Cross out common factors

y=- 2A/2 cos( 2π x/5)