Trigonometry Examples

$\cos (80 + x) \cdot \cos (2 + x) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Rearrange terms.

$\cos (x + 80) \cdot \cos (2 + x) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.2

Double negation

$\cos (x - (- 80)) \cdot \cos (2 + x) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.3

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$(\cos (x) \cos (- 80) + \sin (x) \sin (- 80)) \cdot \cos (2 + x) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.4

Simplify each term.

Tap for more steps...

Step 1.4.1

Evaluate $\cos (- 80)$ .

$(\cos (x) \cdot - 0.11038724 + \sin (x) \sin (- 80)) \cdot \cos (2 + x) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.4.2

Move $- 0.11038724$ to the left of $\cos (x)$ .

$(- 0.11038724 \cdot \cos (x) + \sin (x) \sin (- 80)) \cdot \cos (2 + x) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.4.3

Evaluate $\sin (- 80)$ .

$(- 0.11038724 \cos (x) + \sin (x) \cdot 0.99388865) \cdot \cos (2 + x) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.4.4

Move $0.99388865$ to the left of $\sin (x)$ .

$(- 0.11038724 \cos (x) + 0.99388865 \sin (x)) \cdot \cos (2 + x) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.5

Rearrange terms.

$(- 0.11038724 \cos (x) + 0.99388865 \sin (x)) \cdot \cos (x + 2) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.6

Double negation

$(- 0.11038724 \cos (x) + 0.99388865 \sin (x)) \cdot \cos (x - (- 2)) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.7

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$(- 0.11038724 \cos (x) + 0.99388865 \sin (x)) \cdot (\cos (x) \cos (- 2) + \sin (x) \sin (- 2)) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.8

Simplify each term.

Tap for more steps...

Step 1.8.1

Evaluate $\cos (- 2)$ .

$(- 0.11038724 \cos (x) + 0.99388865 \sin (x)) \cdot (\cos (x) \cdot - 0.41614683 + \sin (x) \sin (- 2)) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.8.2

Move $- 0.41614683$ to the left of $\cos (x)$ .

$(- 0.11038724 \cos (x) + 0.99388865 \sin (x)) \cdot (- 0.41614683 \cdot \cos (x) + \sin (x) \sin (- 2)) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.8.3

Evaluate $\sin (- 2)$ .

$(- 0.11038724 \cos (x) + 0.99388865 \sin (x)) \cdot (- 0.41614683 \cos (x) + \sin (x) \cdot - 0.90929742) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.8.4

Move $- 0.90929742$ to the left of $\sin (x)$ .

$(- 0.11038724 \cos (x) + 0.99388865 \sin (x)) \cdot (- 0.41614683 \cos (x) - 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.9

Expand $(- 0.11038724 \cos (x) + 0.99388865 \sin (x)) (- 0.41614683 \cos (x) - 0.90929742 \sin (x))$ using the FOIL Method.

Tap for more steps...

Step 1.9.1

Apply the distributive property.

$- 0.11038724 \cos (x) (- 0.41614683 \cos (x) - 0.90929742 \sin (x)) + 0.99388865 \sin (x) (- 0.41614683 \cos (x) - 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.9.2

Apply the distributive property.

$- 0.11038724 \cos (x) (- 0.41614683 \cos (x)) - 0.11038724 \cos (x) (- 0.90929742 \sin (x)) + 0.99388865 \sin (x) (- 0.41614683 \cos (x) - 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.9.3

Apply the distributive property.

$- 0.11038724 \cos (x) (- 0.41614683 \cos (x)) - 0.11038724 \cos (x) (- 0.90929742 \sin (x)) + 0.99388865 \sin (x) (- 0.41614683 \cos (x)) + 0.99388865 \sin (x) (- 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.10

Simplify and combine like terms.

Tap for more steps...

Step 1.10.1

Simplify each term.

Tap for more steps...

Step 1.10.1.1

Multiply $- 0.11038724 \cos (x) (- 0.41614683 \cos (x))$ .

Tap for more steps...

Step 1.10.1.1.1

Multiply $- 0.41614683$ by $- 0.11038724$ .

$0.0459373 \cos (x) \cos (x) - 0.11038724 \cos (x) (- 0.90929742 \sin (x)) + 0.99388865 \sin (x) (- 0.41614683 \cos (x)) + 0.99388865 \sin (x) (- 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.10.1.1.2

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

$0.0459373 (\cos^{1} (x) \cos (x)) - 0.11038724 \cos (x) (- 0.90929742 \sin (x)) + 0.99388865 \sin (x) (- 0.41614683 \cos (x)) + 0.99388865 \sin (x) (- 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.10.1.1.3

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

$0.0459373 (\cos^{1} (x) \cos^{1} (x)) - 0.11038724 \cos (x) (- 0.90929742 \sin (x)) + 0.99388865 \sin (x) (- 0.41614683 \cos (x)) + 0.99388865 \sin (x) (- 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.10.1.1.4

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

$0.0459373 {\cos (x)}^{1 + 1} - 0.11038724 \cos (x) (- 0.90929742 \sin (x)) + 0.99388865 \sin (x) (- 0.41614683 \cos (x)) + 0.99388865 \sin (x) (- 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.10.1.1.5

Add $1$ and $1$ .

$0.0459373 \cos^{2} (x) - 0.11038724 \cos (x) (- 0.90929742 \sin (x)) + 0.99388865 \sin (x) (- 0.41614683 \cos (x)) + 0.99388865 \sin (x) (- 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.10.1.2

Multiply $- 0.90929742$ by $- 0.11038724$ .

$0.0459373 \cos^{2} (x) + 0.10037483 \cos (x) \sin (x) + 0.99388865 \sin (x) (- 0.41614683 \cos (x)) + 0.99388865 \sin (x) (- 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.10.1.3

Multiply $- 0.41614683$ by $0.99388865$ .

$0.0459373 \cos^{2} (x) + 0.10037483 \cos (x) \sin (x) - 0.41360361 \sin (x) \cos (x) + 0.99388865 \sin (x) (- 0.90929742 \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.10.1.4

Multiply $0.99388865 \sin (x) (- 0.90929742 \sin (x))$ .

Tap for more steps...

Step 1.10.1.4.1

Multiply $- 0.90929742$ by $0.99388865$ .

$0.0459373 \cos^{2} (x) + 0.10037483 \cos (x) \sin (x) - 0.41360361 \sin (x) \cos (x) - 0.90374039 \sin (x) \sin (x) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.10.1.4.2

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

$0.0459373 \cos^{2} (x) + 0.10037483 \cos (x) \sin (x) - 0.41360361 \sin (x) \cos (x) - 0.90374039 (\sin^{1} (x) \sin (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.10.1.4.3

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

$0.0459373 \cos^{2} (x) + 0.10037483 \cos (x) \sin (x) - 0.41360361 \sin (x) \cos (x) - 0.90374039 (\sin^{1} (x) \sin^{1} (x)) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.10.1.4.4

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

$0.0459373 \cos^{2} (x) + 0.10037483 \cos (x) \sin (x) - 0.41360361 \sin (x) \cos (x) - 0.90374039 {\sin (x)}^{1 + 1} + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.10.1.4.5

Add $1$ and $1$ .

$0.0459373 \cos^{2} (x) + 0.10037483 \cos (x) \sin (x) - 0.41360361 \sin (x) \cos (x) - 0.90374039 \sin^{2} (x) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.10.2

Reorder the factors of $- 0.41360361 \sin (x) \cos (x)$ .

$0.0459373 \cos^{2} (x) + 0.10037483 \cos (x) \sin (x) - 0.41360361 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.10.3

Subtract $0.41360361 \cos (x) \sin (x)$ from $0.10037483 \cos (x) \sin (x)$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + \sin (80 + x) \cdot \sin (20 + x)$

Step 1.11

Rearrange terms.

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + \sin (x + 80) \cdot \sin (20 + x)$

Step 1.12

Double negation

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + \sin (x - (- 80)) \cdot \sin (20 + x)$

Step 1.13

Apply the difference of angles identity.

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (\sin (x) \cos (- 80) - \cos (x) \sin (- 80)) \cdot \sin (20 + x)$

Step 1.14

Simplify each term.

Tap for more steps...

Step 1.14.1

Evaluate $\cos (- 80)$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (\sin (x) \cdot - 0.11038724 - \cos (x) \sin (- 80)) \cdot \sin (20 + x)$

Step 1.14.2

Move $- 0.11038724$ to the left of $\sin (x)$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \cdot \sin (x) - \cos (x) \sin (- 80)) \cdot \sin (20 + x)$

Step 1.14.3

Evaluate $\sin (- 80)$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \sin (x) - \cos (x) \cdot 0.99388865) \cdot \sin (20 + x)$

Step 1.14.4

Multiply $0.99388865$ by $- 1$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \sin (x) - 0.99388865 \cos (x)) \cdot \sin (20 + x)$

Step 1.15

Rearrange terms.

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \sin (x) - 0.99388865 \cos (x)) \cdot \sin (x + 20)$

Step 1.16

Double negation

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \sin (x) - 0.99388865 \cos (x)) \cdot \sin (x - (- 20))$

Step 1.17

Apply the difference of angles identity.

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \sin (x) - 0.99388865 \cos (x)) \cdot (\sin (x) \cos (- 20) - \cos (x) \sin (- 20))$

Step 1.18

Simplify each term.

Tap for more steps...

Step 1.18.1

Evaluate $\cos (- 20)$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \sin (x) - 0.99388865 \cos (x)) \cdot (\sin (x) \cdot 0.40808206 - \cos (x) \sin (- 20))$

Step 1.18.2

Move $0.40808206$ to the left of $\sin (x)$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \sin (x) - 0.99388865 \cos (x)) \cdot (0.40808206 \cdot \sin (x) - \cos (x) \sin (- 20))$

Step 1.18.3

Evaluate $\sin (- 20)$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \sin (x) - 0.99388865 \cos (x)) \cdot (0.40808206 \sin (x) - \cos (x) \cdot - 0.91294525)$

Step 1.18.4

Multiply $- 0.91294525$ by $- 1$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) + (- 0.11038724 \sin (x) - 0.99388865 \cos (x)) \cdot (0.40808206 \sin (x) + 0.91294525 \cos (x))$

Step 1.19

Expand $(- 0.11038724 \sin (x) - 0.99388865 \cos (x)) (0.40808206 \sin (x) + 0.91294525 \cos (x))$ using the FOIL Method.

Tap for more steps...

Step 1.19.1

Apply the distributive property.

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.11038724 \sin (x) (0.40808206 \sin (x) + 0.91294525 \cos (x)) - 0.99388865 \cos (x) (0.40808206 \sin (x) + 0.91294525 \cos (x))$

Step 1.19.2

Apply the distributive property.

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.11038724 \sin (x) (0.40808206 \sin (x)) - 0.11038724 \sin (x) (0.91294525 \cos (x)) - 0.99388865 \cos (x) (0.40808206 \sin (x) + 0.91294525 \cos (x))$

Step 1.19.3

Apply the distributive property.

Step 1.20

Simplify and combine like terms.

Tap for more steps...

Step 1.20.1

Simplify each term.

Tap for more steps...

Step 1.20.1.1

Multiply $- 0.11038724 \sin (x) (0.40808206 \sin (x))$ .

Tap for more steps...

Step 1.20.1.1.1

Multiply $0.40808206$ by $- 0.11038724$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin (x) \sin (x) - 0.11038724 \sin (x) (0.91294525 \cos (x)) - 0.99388865 \cos (x) (0.40808206 \sin (x)) - 0.99388865 \cos (x) (0.91294525 \cos (x))$

Step 1.20.1.1.2

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

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 (\sin^{1} (x) \sin (x)) - 0.11038724 \sin (x) (0.91294525 \cos (x)) - 0.99388865 \cos (x) (0.40808206 \sin (x)) - 0.99388865 \cos (x) (0.91294525 \cos (x))$

Step 1.20.1.1.3

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

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 (\sin^{1} (x) \sin^{1} (x)) - 0.11038724 \sin (x) (0.91294525 \cos (x)) - 0.99388865 \cos (x) (0.40808206 \sin (x)) - 0.99388865 \cos (x) (0.91294525 \cos (x))$

Step 1.20.1.1.4

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

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 {\sin (x)}^{1 + 1} - 0.11038724 \sin (x) (0.91294525 \cos (x)) - 0.99388865 \cos (x) (0.40808206 \sin (x)) - 0.99388865 \cos (x) (0.91294525 \cos (x))$

Step 1.20.1.1.5

Add $1$ and $1$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin^{2} (x) - 0.11038724 \sin (x) (0.91294525 \cos (x)) - 0.99388865 \cos (x) (0.40808206 \sin (x)) - 0.99388865 \cos (x) (0.91294525 \cos (x))$

Step 1.20.1.2

Multiply $0.91294525$ by $- 0.11038724$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin^{2} (x) - 0.10077751 \sin (x) \cos (x) - 0.99388865 \cos (x) (0.40808206 \sin (x)) - 0.99388865 \cos (x) (0.91294525 \cos (x))$

Step 1.20.1.3

Multiply $0.40808206$ by $- 0.99388865$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin^{2} (x) - 0.10077751 \sin (x) \cos (x) - 0.40558813 \cos (x) \sin (x) - 0.99388865 \cos (x) (0.91294525 \cos (x))$

Step 1.20.1.4

Multiply $- 0.99388865 \cos (x) (0.91294525 \cos (x))$ .

Tap for more steps...

Step 1.20.1.4.1

Multiply $0.91294525$ by $- 0.99388865$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin^{2} (x) - 0.10077751 \sin (x) \cos (x) - 0.40558813 \cos (x) \sin (x) - 0.90736592 \cos (x) \cos (x)$

Step 1.20.1.4.2

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

Step 1.20.1.4.3

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

Step 1.20.1.4.4

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

Step 1.20.1.4.5

Add $1$ and $1$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin^{2} (x) - 0.10077751 \sin (x) \cos (x) - 0.40558813 \cos (x) \sin (x) - 0.90736592 \cos^{2} (x)$

Step 1.20.2

Reorder the factors of $- 0.10077751 \sin (x) \cos (x)$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin^{2} (x) - 0.10077751 \cos (x) \sin (x) - 0.40558813 \cos (x) \sin (x) - 0.90736592 \cos^{2} (x)$

Step 1.20.3

Subtract $0.40558813 \cos (x) \sin (x)$ from $- 0.10077751 \cos (x) \sin (x)$ .

$0.0459373 \cos^{2} (x) - 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin^{2} (x) - 0.50636564 \cos (x) \sin (x) - 0.90736592 \cos^{2} (x)$

Step 2

Simplify by adding terms.

Tap for more steps...

Step 2.1

Subtract $0.90736592 \cos^{2} (x)$ from $0.0459373 \cos^{2} (x)$ .

$- 0.31322878 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin^{2} (x) - 0.50636564 \cos (x) \sin (x) - 0.86142862 \cos^{2} (x)$

Step 2.2

Subtract $0.50636564 \cos (x) \sin (x)$ from $- 0.31322878 \cos (x) \sin (x)$ .

$- 0.81959442 \cos (x) \sin (x) - 0.90374039 \sin^{2} (x) - 0.04504705 \sin^{2} (x) - 0.86142862 \cos^{2} (x)$

Step 2.3

Subtract $0.04504705 \sin^{2} (x)$ from $- 0.90374039 \sin^{2} (x)$ .

$- 0.81959442 \cos (x) \sin (x) - 0.94878744 \sin^{2} (x) - 0.86142862 \cos^{2} (x)$