Algebra Examples

${(4 \cos (θ) - 3 \sin (θ))}^{2} + {(3 \cos (θ) + 4 \sin (θ))}^{2}$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Rewrite ${(4 \cos (θ) - 3 \sin (θ))}^{2}$ as $(4 \cos (θ) - 3 \sin (θ)) (4 \cos (θ) - 3 \sin (θ))$ .

$(4 \cos (θ) - 3 \sin (θ)) (4 \cos (θ) - 3 \sin (θ)) + {(3 \cos (θ) + 4 \sin (θ))}^{2}$

Step 1.2

Expand $(4 \cos (θ) - 3 \sin (θ)) (4 \cos (θ) - 3 \sin (θ))$ using the FOIL Method.

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$4 \cos (θ) (4 \cos (θ) - 3 \sin (θ)) - 3 \sin (θ) (4 \cos (θ) - 3 \sin (θ)) + {(3 \cos (θ) + 4 \sin (θ))}^{2}$

Step 1.2.2

Apply the distributive property.

$4 \cos (θ) (4 \cos (θ)) + 4 \cos (θ) (- 3 \sin (θ)) - 3 \sin (θ) (4 \cos (θ) - 3 \sin (θ)) + {(3 \cos (θ) + 4 \sin (θ))}^{2}$

Step 1.2.3

Apply the distributive property.

$4 \cos (θ) (4 \cos (θ)) + 4 \cos (θ) (- 3 \sin (θ)) - 3 \sin (θ) (4 \cos (θ)) - 3 \sin (θ) (- 3 \sin (θ)) + {(3 \cos (θ) + 4 \sin (θ))}^{2}$

Step 1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

Multiply $4 \cos (θ) (4 \cos (θ))$ .

Tap for more steps...

Step 1.3.1.1.1

Multiply $4$ by $4$ .

$16 \cos (θ) \cos (θ) + 4 \cos (θ) (- 3 \sin (θ)) - 3 \sin (θ) (4 \cos (θ)) - 3 \sin (θ) (- 3 \sin (θ)) + {(3 \cos (θ) + 4 \sin (θ))}^{2}$

Step 1.3.1.1.2

Raise $\cos (θ)$ to the power of $1$ .

$16 (\cos^{1} (θ) \cos (θ)) + 4 \cos (θ) (- 3 \sin (θ)) - 3 \sin (θ) (4 \cos (θ)) - 3 \sin (θ) (- 3 \sin (θ)) + {(3 \cos (θ) + 4 \sin (θ))}^{2}$

Step 1.3.1.1.3

Raise $\cos (θ)$ to the power of $1$ .

$16 (\cos^{1} (θ) \cos^{1} (θ)) + 4 \cos (θ) (- 3 \sin (θ)) - 3 \sin (θ) (4 \cos (θ)) - 3 \sin (θ) (- 3 \sin (θ)) + {(3 \cos (θ) + 4 \sin (θ))}^{2}$

Step 1.3.1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$16 {\cos (θ)}^{1 + 1} + 4 \cos (θ) (- 3 \sin (θ)) - 3 \sin (θ) (4 \cos (θ)) - 3 \sin (θ) (- 3 \sin (θ)) + {(3 \cos (θ) + 4 \sin (θ))}^{2}$

Step 1.3.1.1.5

Add $1$ and $1$ .

$16 \cos^{2} (θ) + 4 \cos (θ) (- 3 \sin (θ)) - 3 \sin (θ) (4 \cos (θ)) - 3 \sin (θ) (- 3 \sin (θ)) + {(3 \cos (θ) + 4 \sin (θ))}^{2}$

Step 1.3.1.2

Multiply $- 3$ by $4$ .

$16 \cos^{2} (θ) - 12 \cos (θ) \sin (θ) - 3 \sin (θ) (4 \cos (θ)) - 3 \sin (θ) (- 3 \sin (θ)) + {(3 \cos (θ) + 4 \sin (θ))}^{2}$

Step 1.3.1.3

Multiply $4$ by $- 3$ .

$16 \cos^{2} (θ) - 12 \cos (θ) \sin (θ) - 12 \sin (θ) \cos (θ) - 3 \sin (θ) (- 3 \sin (θ)) + {(3 \cos (θ) + 4 \sin (θ))}^{2}$

Step 1.3.1.4

Multiply $- 3 \sin (θ) (- 3 \sin (θ))$ .

Tap for more steps...

Step 1.3.1.4.1

Multiply $- 3$ by $- 3$ .

$16 \cos^{2} (θ) - 12 \cos (θ) \sin (θ) - 12 \sin (θ) \cos (θ) + 9 \sin (θ) \sin (θ) + {(3 \cos (θ) + 4 \sin (θ))}^{2}$

Step 1.3.1.4.2

Raise $\sin (θ)$ to the power of $1$ .

$16 \cos^{2} (θ) - 12 \cos (θ) \sin (θ) - 12 \sin (θ) \cos (θ) + 9 (\sin^{1} (θ) \sin (θ)) + {(3 \cos (θ) + 4 \sin (θ))}^{2}$

Step 1.3.1.4.3

Raise $\sin (θ)$ to the power of $1$ .

$16 \cos^{2} (θ) - 12 \cos (θ) \sin (θ) - 12 \sin (θ) \cos (θ) + 9 (\sin^{1} (θ) \sin^{1} (θ)) + {(3 \cos (θ) + 4 \sin (θ))}^{2}$

Step 1.3.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$16 \cos^{2} (θ) - 12 \cos (θ) \sin (θ) - 12 \sin (θ) \cos (θ) + 9 {\sin (θ)}^{1 + 1} + {(3 \cos (θ) + 4 \sin (θ))}^{2}$

Step 1.3.1.4.5

Add $1$ and $1$ .

$16 \cos^{2} (θ) - 12 \cos (θ) \sin (θ) - 12 \sin (θ) \cos (θ) + 9 \sin^{2} (θ) + {(3 \cos (θ) + 4 \sin (θ))}^{2}$

Step 1.3.2

Reorder the factors of $- 12 \sin (θ) \cos (θ)$ .

$16 \cos^{2} (θ) - 12 \cos (θ) \sin (θ) - 12 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + {(3 \cos (θ) + 4 \sin (θ))}^{2}$

Step 1.3.3

Subtract $12 \cos (θ) \sin (θ)$ from $- 12 \cos (θ) \sin (θ)$ .

$16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + {(3 \cos (θ) + 4 \sin (θ))}^{2}$

Step 1.4

Rewrite ${(3 \cos (θ) + 4 \sin (θ))}^{2}$ as $(3 \cos (θ) + 4 \sin (θ)) (3 \cos (θ) + 4 \sin (θ))$ .

$16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + (3 \cos (θ) + 4 \sin (θ)) (3 \cos (θ) + 4 \sin (θ))$

Step 1.5

Expand $(3 \cos (θ) + 4 \sin (θ)) (3 \cos (θ) + 4 \sin (θ))$ using the FOIL Method.

Tap for more steps...

Step 1.5.1

Apply the distributive property.

$16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + 3 \cos (θ) (3 \cos (θ) + 4 \sin (θ)) + 4 \sin (θ) (3 \cos (θ) + 4 \sin (θ))$

Step 1.5.2

Apply the distributive property.

$16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + 3 \cos (θ) (3 \cos (θ)) + 3 \cos (θ) (4 \sin (θ)) + 4 \sin (θ) (3 \cos (θ) + 4 \sin (θ))$

Step 1.5.3

Apply the distributive property.

$16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + 3 \cos (θ) (3 \cos (θ)) + 3 \cos (θ) (4 \sin (θ)) + 4 \sin (θ) (3 \cos (θ)) + 4 \sin (θ) (4 \sin (θ))$

Step 1.6

Simplify and combine like terms.

Tap for more steps...

Step 1.6.1

Simplify each term.

Tap for more steps...

Step 1.6.1.1

Multiply $3 \cos (θ) (3 \cos (θ))$ .

Tap for more steps...

Step 1.6.1.1.1

Multiply $3$ by $3$ .

$16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + 9 \cos (θ) \cos (θ) + 3 \cos (θ) (4 \sin (θ)) + 4 \sin (θ) (3 \cos (θ)) + 4 \sin (θ) (4 \sin (θ))$

Step 1.6.1.1.2

Raise $\cos (θ)$ to the power of $1$ .

$16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + 9 (\cos^{1} (θ) \cos (θ)) + 3 \cos (θ) (4 \sin (θ)) + 4 \sin (θ) (3 \cos (θ)) + 4 \sin (θ) (4 \sin (θ))$

Step 1.6.1.1.3

Raise $\cos (θ)$ to the power of $1$ .

$16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + 9 (\cos^{1} (θ) \cos^{1} (θ)) + 3 \cos (θ) (4 \sin (θ)) + 4 \sin (θ) (3 \cos (θ)) + 4 \sin (θ) (4 \sin (θ))$

Step 1.6.1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + 9 {\cos (θ)}^{1 + 1} + 3 \cos (θ) (4 \sin (θ)) + 4 \sin (θ) (3 \cos (θ)) + 4 \sin (θ) (4 \sin (θ))$

Step 1.6.1.1.5

Add $1$ and $1$ .

$16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + 9 \cos^{2} (θ) + 3 \cos (θ) (4 \sin (θ)) + 4 \sin (θ) (3 \cos (θ)) + 4 \sin (θ) (4 \sin (θ))$

Step 1.6.1.2

Multiply $4$ by $3$ .

$16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + 9 \cos^{2} (θ) + 12 \cos (θ) \sin (θ) + 4 \sin (θ) (3 \cos (θ)) + 4 \sin (θ) (4 \sin (θ))$

Step 1.6.1.3

Multiply $3$ by $4$ .

$16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + 9 \cos^{2} (θ) + 12 \cos (θ) \sin (θ) + 12 \sin (θ) \cos (θ) + 4 \sin (θ) (4 \sin (θ))$

Step 1.6.1.4

Multiply $4 \sin (θ) (4 \sin (θ))$ .

Tap for more steps...

Step 1.6.1.4.1

Multiply $4$ by $4$ .

$16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + 9 \cos^{2} (θ) + 12 \cos (θ) \sin (θ) + 12 \sin (θ) \cos (θ) + 16 \sin (θ) \sin (θ)$

Step 1.6.1.4.2

Raise $\sin (θ)$ to the power of $1$ .

$16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + 9 \cos^{2} (θ) + 12 \cos (θ) \sin (θ) + 12 \sin (θ) \cos (θ) + 16 (\sin^{1} (θ) \sin (θ))$

Step 1.6.1.4.3

Raise $\sin (θ)$ to the power of $1$ .

$16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + 9 \cos^{2} (θ) + 12 \cos (θ) \sin (θ) + 12 \sin (θ) \cos (θ) + 16 (\sin^{1} (θ) \sin^{1} (θ))$

Step 1.6.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + 9 \cos^{2} (θ) + 12 \cos (θ) \sin (θ) + 12 \sin (θ) \cos (θ) + 16 {\sin (θ)}^{1 + 1}$

Step 1.6.1.4.5

Add $1$ and $1$ .

$16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + 9 \cos^{2} (θ) + 12 \cos (θ) \sin (θ) + 12 \sin (θ) \cos (θ) + 16 \sin^{2} (θ)$

Step 1.6.2

Reorder the factors of $12 \sin (θ) \cos (θ)$ .

$16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + 9 \cos^{2} (θ) + 12 \cos (θ) \sin (θ) + 12 \cos (θ) \sin (θ) + 16 \sin^{2} (θ)$

Step 1.6.3

Add $12 \cos (θ) \sin (θ)$ and $12 \cos (θ) \sin (θ)$ .

$16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + 9 \cos^{2} (θ) + 24 \cos (θ) \sin (θ) + 16 \sin^{2} (θ)$

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Combine the opposite terms in $16 \cos^{2} (θ) - 24 \cos (θ) \sin (θ) + 9 \sin^{2} (θ) + 9 \cos^{2} (θ) + 24 \cos (θ) \sin (θ) + 16 \sin^{2} (θ)$ .

Tap for more steps...

Step 2.1.1

Add $- 24 \cos (θ) \sin (θ)$ and $24 \cos (θ) \sin (θ)$ .

$16 \cos^{2} (θ) + 0 + 9 \sin^{2} (θ) + 9 \cos^{2} (θ) + 16 \sin^{2} (θ)$

Step 2.1.2

Add $16 \cos^{2} (θ)$ and $0$ .

$16 \cos^{2} (θ) + 9 \sin^{2} (θ) + 9 \cos^{2} (θ) + 16 \sin^{2} (θ)$

Step 2.2

Move $16 \sin^{2} (θ)$ .

$16 \cos^{2} (θ) + 16 \sin^{2} (θ) + 9 \sin^{2} (θ) + 9 \cos^{2} (θ)$

Step 2.3

Factor $16$ out of $16 \cos^{2} (θ)$ .

$16 (\cos^{2} (θ)) + 16 \sin^{2} (θ) + 9 \sin^{2} (θ) + 9 \cos^{2} (θ)$

Step 2.4

Factor $16$ out of $16 \sin^{2} (θ)$ .

$16 (\cos^{2} (θ)) + 16 (\sin^{2} (θ)) + 9 \sin^{2} (θ) + 9 \cos^{2} (θ)$

Step 2.5

Factor $16$ out of $16 (\cos^{2} (θ)) + 16 (\sin^{2} (θ))$ .

$16 (\cos^{2} (θ) + \sin^{2} (θ)) + 9 \sin^{2} (θ) + 9 \cos^{2} (θ)$

Step 3

Rearrange terms.

$16 (\sin^{2} (θ) + \cos^{2} (θ)) + 9 \sin^{2} (θ) + 9 \cos^{2} (θ)$

Step 4

Apply pythagorean identity.

$16 \cdot 1 + 9 \sin^{2} (θ) + 9 \cos^{2} (θ)$

Step 5

Simplify with factoring out.

Tap for more steps...

Step 5.1

Factor $9$ out of $9 \sin^{2} (θ)$ .

$16 \cdot 1 + 9 (\sin^{2} (θ)) + 9 \cos^{2} (θ)$

Step 5.2

Factor $9$ out of $9 \cos^{2} (θ)$ .

$16 \cdot 1 + 9 (\sin^{2} (θ)) + 9 (\cos^{2} (θ))$

Step 5.3

Factor $9$ out of $9 (\sin^{2} (θ)) + 9 (\cos^{2} (θ))$ .

$16 \cdot 1 + 9 (\sin^{2} (θ) + \cos^{2} (θ))$

Step 6

Apply pythagorean identity.

$16 \cdot 1 + 9 \cdot 1$

Step 7

Simplify terms.

Tap for more steps...

Step 7.1

Simplify each term.

Tap for more steps...

Step 7.1.1

Multiply $16$ by $1$ .

$16 + 9 \cdot 1$

Step 7.1.2

Multiply $9$ by $1$ .

$16 + 9$

Step 7.2

Add $16$ and $9$ .

$25$