We are given the two series below, where one of them represents sin x and the other one represents cos x.
a. x - x^3/3! + x^5/5! - x^7/7! + ⋯ [0.2cm]
b. 1 - x^2/2! + x^4/4! - x^6/6! + ⋯
If we substitute x=0 into each series, we will see that the first one will be equal to 0 while the second one will be 1.
a. 0 - 0^3/3! + 0^5/5! - 0^7/7! + ⋯ = 0 [0.2cm]
b. 1 - 0^2/2! + 0^4/4! - 0^6/6! + ⋯ = 1
Now, let's find the values that sine and cosine approach as the angle measure approaches zero.
From the diagram above, we can see that as m ∠ A approaches zero, the length of the opposite side (BC) also approaches zero. In consequence, sin A approaches zero.
sin A = BC/AB
⇓
sin A approaches 0 asm∠ A tends to 0
Since the first series (
a.) tends to zero as x approaches zero, we conclude that the first series represents sin x.
sin x = x - x^3/3! + x^5/5! - x^7/7! + ⋯
Additionally, from the diagram, we can also see that as m ∠ A approaches zero, the length of the hypotenuse (AB) approaches the same length as AC. In consequence, cos A approaches 1.
cos A = AC/AB
⇓
cos A approaches 1 asm∠ A tends to 0
Thus, we conclude that the second series (
b.) represents cos x.
cos x = 1 - x^2/2! + x^4/4! - x^6/6! + ⋯
Finding Sine and Cosine of π6
Let's find the sine of π6 by using the series.
sin x = x - x^3/3! + x^5/5! - x^7/7! + ⋯
sin π/6 = π/6 - ( π/6)^3/3! + ( π/6)^5/5! - ( π/6)^7/7! + ⋯
sin π/6 = π/6 - π^3/216/3! + π^5/7776/5! - π^7/279 936/7! + ⋯
sin π/6 = π/6 - π^3/126* 3! + π^5/7776* 5! - π^7/279 936* 7! + ⋯
sin π/6 = π/6 - π^3/1296 + π^5/933 120 - π^7/1 410 877 440 + ⋯
sin π/6 ≈ 3.1415/6 - 3.1415^3/1296 + 3.1415^5/933 120 - 3.1415^7/1 410 877 440 + ⋯
sin π/6 ≈ 3.1415/6 - 31.006/1296 + 306.02/933 120 - 3020.29/1 410 877 440 + ⋯
sin π/6 ≈ 0.523 - 0.023 + 0.000 33 - 0.000 002
sin π/6 ≈ 0.500 33
By rounding to one decimal place, we have that sin π6≈ 0.5. Similarly, let's find cosine of π6 by using its series.
cos x = 1 - x^2/2! + x^4/4! - x^6/6! + ⋯
cos π/6 = 1- ( π/6)^2/2! + ( π/6)^4/4! - ( π/6)^6/6! + ⋯
cos π/6 = 1 - π^2/36/2! + π^4/1296/4! - π^6/46 656/6! + ⋯
cos π/6 = 1 - π^2/36* 2! + π^4/1296* 4! - π^6/46 656* 6! + ⋯
cos π/6 = 1 - π^2/72 + π^4/31 104 - π^6/33 592 320 + ⋯
cos π/6 ≈ 1 - 3.1415^2/72 + 3.1415^4/31 104 - 3.1415^6/33 592 320 + ⋯
cos π/6 ≈ 1 - 9.869/72 + 97.409/31 104 - 961.389/33 592 320 + ⋯
cos π/6 ≈ 1 - 0.137 + 0.0031 - 0.000 028
cos π/6 ≈ 0.866
Consequently, we have that cos π6≈ 0.866. Notice that, using the fact that π=180^(∘), we obtain π6=30^(∘). This makes sense, since in this chapter, we learned the two relations below.
sin 30^(∘) = 0.5 cos 30^(∘) ≈ 0.866
As we can notice, the values above coincide with the values obtained using the series.
