3. Sum and Difference of Angles Identities - 13. Trigonometric Identities and Equations

a We are given two functions modeling different signals.

First signal: & y=20sin(3θ+45^(∘)) Second signal: & y=20sin(3θ+225^(∘)) First we want to find the sum of these two functions. Before we do so, let's simplify each one of them by using one of the Sum and Difference Identities describing the sine of the sum of two angles. sin(A+B)=sin Acos B + cos Asin B We will now use this identity to simplify the first function.

y=20sin(3θ+45^(∘))

Substitute

sin(3θ+45^(∘))= sin3θcos45^(∘)+cos3θsin45^(∘)

y=20( sin3θcos45^(∘)+cos3θsin45^(∘))

▼

Simplify right-hand side

Distr

Distribute 20

y=20sin3θcos45^(∘)+20cos3θsin45^(∘)

\ifnumequal{45}{0}{\cos\left(0^\circ\right)=1}{}\ifnumequal{45}{30}{\cos\left(30^\circ\right)=\dfrac{\sqrt{3}}{2}}{}\ifnumequal{45}{45}{\cos\left(45^\circ\right)=\dfrac{\sqrt{2}}{2}}{}\ifnumequal{45}{60}{\cos\left(60^\circ\right)=\dfrac{1}{2}}{}\ifnumequal{45}{90}{\cos\left(90^\circ\right)=0}{}\ifnumequal{45}{120}{\cos\left(120^\circ\right)=\text{-} \dfrac{1}{2}}{}\ifnumequal{45}{135}{\cos\left(135^\circ\right)=\text{-} \dfrac{\sqrt{2}}{2}}{}\ifnumequal{45}{150}{\cos\left(150^\circ\right)=\text{-} \dfrac{\sqrt{3}}{2}}{}\ifnumequal{45}{180}{\cos\left(180^\circ\right)=\text{-} 1}{}\ifnumequal{45}{210}{\cos\left(210^\circ\right)=\text{-} \dfrac{\sqrt 3}2}{}\ifnumequal{45}{225}{\cos\left(225^\circ\right)=\text{-} \dfrac {\sqrt{2}} {2}}{}\ifnumequal{45}{240}{\cos\left(240^\circ\right)=\text{-} \dfrac {1}2}{}\ifnumequal{45}{270}{\cos\left(270^\circ\right)=0}{}\ifnumequal{45}{300}{\cos\left(300^\circ\right)=\dfrac{1}2}{}\ifnumequal{45}{315}{\cos\left(315^\circ\right)=\dfrac {\sqrt{2}} {2}}{}\ifnumequal{45}{330}{\cos\left(330^\circ\right)=\dfrac{\sqrt 3}2}{}\ifnumequal{45}{360}{\cos\left(360^\circ\right)=1}{}

y=20sin3θ(sqrt(2)/2)+20cos3θsin45^(∘)

\ifnumequal{45}{0}{\sin\left(0^\circ\right)=0}{}\ifnumequal{45}{30}{\sin\left(30^\circ\right)=\dfrac{1}{2}}{}\ifnumequal{45}{45}{\sin\left(45^\circ\right)=\dfrac{\sqrt{2}}{2}}{}\ifnumequal{45}{60}{\sin\left(60^\circ\right)=\dfrac{\sqrt{3}}{2}}{}\ifnumequal{45}{90}{\sin\left(90^\circ\right)=1}{}\ifnumequal{45}{120}{\sin\left(120^\circ\right)=\dfrac{\sqrt{3}}{2}}{}\ifnumequal{45}{135}{\sin\left(135^\circ\right)=\dfrac{\sqrt{2}}{2}}{}\ifnumequal{45}{150}{\sin\left(150^\circ\right)=\dfrac{1}{2}}{}\ifnumequal{45}{180}{\sin\left(180^\circ\right)=0}{}\ifnumequal{45}{210}{\sin\left(210^\circ\right)=\text{-} \dfrac 1 2}{}\ifnumequal{45}{225}{\sin\left(225^\circ\right)=\text{-} \dfrac {\sqrt{2}} {2}}{}\ifnumequal{45}{240}{\sin\left(240^\circ\right)=\text{-} \dfrac {\sqrt 3}2}{}\ifnumequal{45}{270}{\sin\left(270^\circ\right)=\text{-}1}{}\ifnumequal{45}{300}{\sin\left(300^\circ\right)=\text{-}\dfrac {\sqrt 3}2}{}\ifnumequal{45}{315}{\sin\left(315^\circ\right)=\text{-} \dfrac {\sqrt{2}} {2}}{}\ifnumequal{45}{330}{\sin\left(330^\circ\right)=\text{-} \dfrac 1 2}{}\ifnumequal{45}{360}{\sin\left(360^\circ\right)=0}{}

y=20sin3θ(sqrt(2)/2)+20cos3θ(sqrt(2)/2)

CommutativePropMult

Commutative Property of Multiplication

y=20(sqrt(2)/2)sin3θ+20(sqrt(2)/2)cos3θ

MoveLeftFacToNum

a*b/c= a* b/c

y=20sqrt(2)/2sin3θ+20sqrt(2)/2cos3θ

CalcQuot

Calculate quotient

y=10sqrt(2)sin3θ+10sqrt(2)cos3θ

Let's now use the same identity to simplify the second function.

y=20sin(3θ+45^(∘))

Substitute

sin(3θ+225^(∘))= sin3θcos225^(∘)+cos3θsin225^(∘)

y=20( sin3θcos225^(∘)+cos3θsin225^(∘))

▼

Simplify right-hand side

Distr

Distribute 20

y=20sin3θcos225^(∘)+20cos3θsin225^(∘)

\ifnumequal{225}{0}{\cos\left(0^\circ\right)=1}{}\ifnumequal{225}{30}{\cos\left(30^\circ\right)=\dfrac{\sqrt{3}}{2}}{}\ifnumequal{225}{45}{\cos\left(45^\circ\right)=\dfrac{\sqrt{2}}{2}}{}\ifnumequal{225}{60}{\cos\left(60^\circ\right)=\dfrac{1}{2}}{}\ifnumequal{225}{90}{\cos\left(90^\circ\right)=0}{}\ifnumequal{225}{120}{\cos\left(120^\circ\right)=\text{-} \dfrac{1}{2}}{}\ifnumequal{225}{135}{\cos\left(135^\circ\right)=\text{-} \dfrac{\sqrt{2}}{2}}{}\ifnumequal{225}{150}{\cos\left(150^\circ\right)=\text{-} \dfrac{\sqrt{3}}{2}}{}\ifnumequal{225}{180}{\cos\left(180^\circ\right)=\text{-} 1}{}\ifnumequal{225}{210}{\cos\left(210^\circ\right)=\text{-} \dfrac{\sqrt 3}2}{}\ifnumequal{225}{225}{\cos\left(225^\circ\right)=\text{-} \dfrac {\sqrt{2}} {2}}{}\ifnumequal{225}{240}{\cos\left(240^\circ\right)=\text{-} \dfrac {1}2}{}\ifnumequal{225}{270}{\cos\left(270^\circ\right)=0}{}\ifnumequal{225}{300}{\cos\left(300^\circ\right)=\dfrac{1}2}{}\ifnumequal{225}{315}{\cos\left(315^\circ\right)=\dfrac {\sqrt{2}} {2}}{}\ifnumequal{225}{330}{\cos\left(330^\circ\right)=\dfrac{\sqrt 3}2}{}\ifnumequal{225}{360}{\cos\left(360^\circ\right)=1}{}

y=20sin3θ(-sqrt(2)/2)+20cos3θsin225^(∘)

\ifnumequal{225}{0}{\sin\left(0^\circ\right)=0}{}\ifnumequal{225}{30}{\sin\left(30^\circ\right)=\dfrac{1}{2}}{}\ifnumequal{225}{45}{\sin\left(45^\circ\right)=\dfrac{\sqrt{2}}{2}}{}\ifnumequal{225}{60}{\sin\left(60^\circ\right)=\dfrac{\sqrt{3}}{2}}{}\ifnumequal{225}{90}{\sin\left(90^\circ\right)=1}{}\ifnumequal{225}{120}{\sin\left(120^\circ\right)=\dfrac{\sqrt{3}}{2}}{}\ifnumequal{225}{135}{\sin\left(135^\circ\right)=\dfrac{\sqrt{2}}{2}}{}\ifnumequal{225}{150}{\sin\left(150^\circ\right)=\dfrac{1}{2}}{}\ifnumequal{225}{180}{\sin\left(180^\circ\right)=0}{}\ifnumequal{225}{210}{\sin\left(210^\circ\right)=\text{-} \dfrac 1 2}{}\ifnumequal{225}{225}{\sin\left(225^\circ\right)=\text{-} \dfrac {\sqrt{2}} {2}}{}\ifnumequal{225}{240}{\sin\left(240^\circ\right)=\text{-} \dfrac {\sqrt 3}2}{}\ifnumequal{225}{270}{\sin\left(270^\circ\right)=\text{-}1}{}\ifnumequal{225}{300}{\sin\left(300^\circ\right)=\text{-}\dfrac {\sqrt 3}2}{}\ifnumequal{225}{315}{\sin\left(315^\circ\right)=\text{-} \dfrac {\sqrt{2}} {2}}{}\ifnumequal{225}{330}{\sin\left(330^\circ\right)=\text{-} \dfrac 1 2}{}\ifnumequal{225}{360}{\sin\left(360^\circ\right)=0}{}

y=20sin3θ(-sqrt(2)/2)+20cos3θ(-sqrt(2)/2)

CommutativePropMult

Commutative Property of Multiplication

y=20(-sqrt(2)/2)sin3θ+20(-sqrt(2)/2)cos3θ

MoveLeftFacToNum

a*b/c= a* b/c

y=-20sqrt(2)/2sin3θ+(-20sqrt(2)/2cos3θ)

CalcQuot

Calculate quotient

y=-10sqrt(2)sin3θ+(-10sqrt(2)cos3θ)

AddNeg

a+(- b)=a-b