a We are given two functions modeling different signals.

First signal : Second signal : y = 20 sin (3 θ + 4 5^{\circ}) y = 20 sin (3 θ + 22 5^{\circ})

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 A cos B + cos A sin B

We will now use this identity to simplify the first function.

y = 20 sin (3 θ + 4 5^{\circ})

Substitute

$sin (3 θ + 4 5^{\circ}) = s i n 3 θ c o s 4 5^{\circ} + c o s 3 θ s i n 4 5^{\circ}$

y = 20 (s i n 3 θ c o s 4 5^{\circ} + c o s 3 θ s i n 4 5^{\circ})

▼

Simplify right-hand side

Distr

Distribute $20$

y = 20 sin 3 θ cos 4 5^{\circ} + 20 cos 3 θ sin 4 5^{\circ}

y = 20 sin 3 θ (\frac{2}{2}) + 20 cos 3 θ sin 4 5^{\circ}

y = 20 sin 3 θ (\frac{2}{2}) + 20 cos 3 θ (\frac{2}{2})

CommutativePropMult

Commutative Property of Multiplication

y = 20 (\frac{2}{2}) sin 3 θ + 20 (\frac{2}{2}) cos 3 θ

MoveLeftFacToNum

$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

y = \frac{2 0 2}{2} sin 3 θ + \frac{2 0 2}{2} cos 3 θ

CalcQuot

Calculate quotient

y = 102 sin 3 θ + 102 cos 3 θ

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

y = 20 sin (3 θ + 4 5^{\circ})

Substitute

$sin (3 θ + 22 5^{\circ}) = s i n 3 θ c o s 22 5^{\circ} + c o s 3 θ s i n 22 5^{\circ}$

y = 20 (s i n 3 θ c o s 22 5^{\circ} + c o s 3 θ s i n 22 5^{\circ})

▼

Simplify right-hand side

Distr

Distribute $20$

y = 20 sin 3 θ cos 22 5^{\circ} + 20 cos 3 θ sin 22 5^{\circ}

y = 20 sin 3 θ (- \frac{2}{2}) + 20 cos 3 θ sin 22 5^{\circ}

y = 20 sin 3 θ (- \frac{2}{2}) + 20 cos 3 θ (- \frac{2}{2})

CommutativePropMult

Commutative Property of Multiplication

y = 20 (- \frac{2}{2}) sin 3 θ + 20 (- \frac{2}{2}) cos 3 θ

MoveLeftFacToNum

$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

y = - \frac{2 0 2}{2} sin 3 θ + (- \frac{2 0 2}{2} cos 3 θ)

CalcQuot

Calculate quotient

y = - 102 sin 3 θ + (- 102 cos 3 θ)

AddNeg

$a + (- b) = a - b$

y = - 102 sin 3 θ - 102 cos 3 θ

Now after we have simplified both functions we can add them.

(102 sin 3 θ + 102 cos 3 θ) + (- 102 sin 3 θ - 102 cos 3 θ)

▼

Simplify

FactorOut

Factor out $sin 3 θ$

sin 3 θ (102 - 102) + 102 cos 3 θ - 102 cos 3 θ

FactorOut

Factor out $cos 3 θ$

sin 3 θ (102 - 102) + cos 3 θ (102 - 102)

DiffZero

$102 - 102 = 0$

sin 3 θ (0) + 102 cos 3 θ (0)

ZeroPropMult

Zero Property of Multiplication

0 + 0

AddTerms

Add terms

0

The sum of the given functions is

0 .

Exercise

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