Trigonometry Examples

$(1, \frac{5}{3})$ , $(1, - 8)$

Step 1

Use the dot product formula to find the angle between two vectors.

$θ = \arccos (\frac{a⃗ \cdot b⃗}{| a⃗ | | b⃗ |})$

Step 2

Find the dot product.

Tap for more steps...

Step 2.1

The dot product of two vectors is the sum of the products of the their components.

$a⃗ \cdot b⃗ = 1 \cdot 1 + \frac{5}{3} \cdot - 8$

Step 2.2

Simplify.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Multiply $1$ by $1$ .

$a⃗ \cdot b⃗ = 1 + \frac{5}{3} \cdot - 8$

Step 2.2.1.2

Multiply $\frac{5}{3} \cdot - 8$ .

Tap for more steps...

Step 2.2.1.2.1

Combine $\frac{5}{3}$ and $- 8$ .

$a⃗ \cdot b⃗ = 1 + \frac{5 \cdot - 8}{3}$

Step 2.2.1.2.2

Multiply $5$ by $- 8$ .

$a⃗ \cdot b⃗ = 1 + \frac{- 40}{3}$

Step 2.2.1.3

Move the negative in front of the fraction.

$a⃗ \cdot b⃗ = 1 - \frac{40}{3}$

Step 2.2.2

Write $1$ as a fraction with a common denominator.

$a⃗ \cdot b⃗ = \frac{3}{3} - \frac{40}{3}$

Step 2.2.3

Combine the numerators over the common denominator.

$a⃗ \cdot b⃗ = \frac{3 - 40}{3}$

Step 2.2.4

Subtract $40$ from $3$ .

$a⃗ \cdot b⃗ = \frac{- 37}{3}$

Step 2.2.5

Move the negative in front of the fraction.

$a⃗ \cdot b⃗ = - \frac{37}{3}$

Step 3

Find the magnitude of $a⃗$ .

Tap for more steps...

Step 3.1

The norm is the square root of the sum of squares of each element in the vector.

$| a⃗ | = \sqrt{1^{2} + {(\frac{5}{3})}^{2}}$

Step 3.2

Simplify.

Tap for more steps...

Step 3.2.1

One to any power is one.

$| a⃗ | = \sqrt{1 + {(\frac{5}{3})}^{2}}$

Step 3.2.2

Apply the product rule to $\frac{5}{3}$ .

$| a⃗ | = \sqrt{1 + \frac{5^{2}}{3^{2}}}$

Step 3.2.3

Raise $5$ to the power of $2$ .

$| a⃗ | = \sqrt{1 + \frac{25}{3^{2}}}$

Step 3.2.4

Raise $3$ to the power of $2$ .

$| a⃗ | = \sqrt{1 + \frac{25}{9}}$

Step 3.2.5

Write $1$ as a fraction with a common denominator.

$| a⃗ | = \sqrt{\frac{9}{9} + \frac{25}{9}}$

Step 3.2.6

Combine the numerators over the common denominator.

$| a⃗ | = \sqrt{\frac{9 + 25}{9}}$

Step 3.2.7

Add $9$ and $25$ .

$| a⃗ | = \sqrt{\frac{34}{9}}$

Step 3.2.8

Rewrite $\sqrt{\frac{34}{9}}$ as $\frac{\sqrt{34}}{\sqrt{9}}$ .

$| a⃗ | = \frac{\sqrt{34}}{\sqrt{9}}$

Step 3.2.9

Simplify the denominator.

Tap for more steps...

Step 3.2.9.1

Rewrite $9$ as $3^{2}$ .

$| a⃗ | = \frac{\sqrt{34}}{\sqrt{3^{2}}}$

Step 3.2.9.2

Pull terms out from under the radical, assuming positive real numbers.

$| a⃗ | = \frac{\sqrt{34}}{3}$

Step 4

Find the magnitude of $b⃗$ .

Tap for more steps...

Step 4.1

The norm is the square root of the sum of squares of each element in the vector.

$| b⃗ | = \sqrt{1^{2} + {(- 8)}^{2}}$

Step 4.2

Simplify.

Tap for more steps...

Step 4.2.1

One to any power is one.

$| b⃗ | = \sqrt{1 + {(- 8)}^{2}}$

Step 4.2.2

Raise $- 8$ to the power of $2$ .

$| b⃗ | = \sqrt{1 + 64}$

Step 4.2.3

Add $1$ and $64$ .

$| b⃗ | = \sqrt{65}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{- \frac{37}{3}}{\frac{\sqrt{34}}{3} \sqrt{65}})$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \arccos (- \frac{37}{3} \cdot \frac{1}{\frac{\sqrt{34}}{3} \sqrt{65}})$

Step 6.2

Combine $\frac{\sqrt{34}}{3}$ and $\sqrt{65}$ .

$θ = \arccos (- \frac{37}{3} \cdot \frac{1}{\frac{\sqrt{34} \sqrt{65}}{3}})$

Step 6.3

Simplify the numerator.

Tap for more steps...

Step 6.3.1

Combine using the product rule for radicals.

$θ = \arccos (- \frac{37}{3} \cdot \frac{1}{\frac{\sqrt{34 \cdot 65}}{3}})$

Step 6.3.2

Multiply $34$ by $65$ .

$θ = \arccos (- \frac{37}{3} \cdot \frac{1}{\frac{\sqrt{2210}}{3}})$

Step 6.4

Multiply the numerator by the reciprocal of the denominator.

$θ = \arccos (- \frac{37}{3} (1 \frac{3}{\sqrt{2210}}))$

Step 6.5

Multiply $\frac{3}{\sqrt{2210}}$ by $1$ .

$θ = \arccos (- \frac{37}{3} \cdot \frac{3}{\sqrt{2210}})$

Step 6.6

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.6.1

Move the leading negative in $- \frac{37}{3}$ into the numerator.

$θ = \arccos (\frac{- 37}{3} \cdot \frac{3}{\sqrt{2210}})$

Step 6.6.2

Cancel the common factor.

$θ = \arccos (\frac{- 37}{3} \cdot \frac{3}{\sqrt{2210}})$

Step 6.6.3

Rewrite the expression.

$θ = \arccos (- 37 \frac{1}{\sqrt{2210}})$

Step 6.7

Combine $- 37$ and $\frac{1}{\sqrt{2210}}$ .

$θ = \arccos (\frac{- 37}{\sqrt{2210}})$

Step 6.8

Move the negative in front of the fraction.

$θ = \arccos (- \frac{37}{\sqrt{2210}})$

Step 6.9

Multiply $\frac{37}{\sqrt{2210}}$ by $\frac{\sqrt{2210}}{\sqrt{2210}}$ .

$θ = \arccos (- (\frac{37}{\sqrt{2210}} \cdot \frac{\sqrt{2210}}{\sqrt{2210}}))$

Step 6.10

Combine and simplify the denominator.

Tap for more steps...

Step 6.10.1

Multiply $\frac{37}{\sqrt{2210}}$ by $\frac{\sqrt{2210}}{\sqrt{2210}}$ .

$θ = \arccos (- \frac{37 \sqrt{2210}}{\sqrt{2210} \sqrt{2210}})$

Step 6.10.2

Raise $\sqrt{2210}$ to the power of $1$ .

$θ = \arccos (- \frac{37 \sqrt{2210}}{{\sqrt{2210}}^{1} \sqrt{2210}})$

Step 6.10.3

Raise $\sqrt{2210}$ to the power of $1$ .

$θ = \arccos (- \frac{37 \sqrt{2210}}{{\sqrt{2210}}^{1} {\sqrt{2210}}^{1}})$

Step 6.10.4

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

$θ = \arccos (- \frac{37 \sqrt{2210}}{{\sqrt{2210}}^{1 + 1}})$

Step 6.10.5

Add $1$ and $1$ .

$θ = \arccos (- \frac{37 \sqrt{2210}}{{\sqrt{2210}}^{2}})$

Step 6.10.6

Rewrite ${\sqrt{2210}}^{2}$ as $2210$ .

Tap for more steps...

Step 6.10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2210}$ as $2210^{\frac{1}{2}}$ .

$θ = \arccos (- \frac{37 \sqrt{2210}}{{(2210^{\frac{1}{2}})}^{2}})$

Step 6.10.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$θ = \arccos (- \frac{37 \sqrt{2210}}{2210^{\frac{1}{2} \cdot 2}})$

Step 6.10.6.3

Combine $\frac{1}{2}$ and $2$ .

$θ = \arccos (- \frac{37 \sqrt{2210}}{2210^{\frac{2}{2}}})$

Step 6.10.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.10.6.4.1

Cancel the common factor.

$θ = \arccos (- \frac{37 \sqrt{2210}}{2210^{\frac{2}{2}}})$

Step 6.10.6.4.2

Rewrite the expression.

$θ = \arccos (- \frac{37 \sqrt{2210}}{2210^{1}})$

Step 6.10.6.5

Evaluate the exponent.

$θ = \arccos (- \frac{37 \sqrt{2210}}{2210})$

Step 6.11

Evaluate $\arccos (- \frac{37 \sqrt{2210}}{2210})$ .

$θ = 141.91122711$