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

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