Trigonometry Examples

(1, \frac{5}{3})

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

(1, - 8)

$(1, - 8)$

Step 1

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

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

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

Step 2

Find the dot product.

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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

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

Step 2.2

Simplify.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply

1

$1$ by

1

$1$ .

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

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

Step 2.2.1.2

Multiply

\frac{5}{3} \cdot - 8

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

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Step 2.2.1.2.1

Combine

\frac{5}{3}

$\frac{5}{3}$ and

- 8

$- 8$ .

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

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

Step 2.2.1.2.2

Multiply

5

$5$ by

- 8

$- 8$ .

a⃗ \cdot b⃗ = 1 + \frac{- 40}{3}

$a⃗ \cdot b⃗ = 1 + \frac{- 40}{3}$

a⃗ \cdot b⃗ = 1 + \frac{- 40}{3}

$a⃗ \cdot b⃗ = 1 + \frac{- 40}{3}$

Step 2.2.1.3

Move the negative in front of the fraction.

a⃗ \cdot b⃗ = 1 - \frac{40}{3}

$a⃗ \cdot b⃗ = 1 - \frac{40}{3}$

a⃗ \cdot b⃗ = 1 - \frac{40}{3}

$a⃗ \cdot b⃗ = 1 - \frac{40}{3}$

Step 2.2.2

Write

1

$1$ as a fraction with a common denominator.

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

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

Step 2.2.3

Combine the numerators over the common denominator.

a⃗ \cdot b⃗ = \frac{3 - 40}{3}

$a⃗ \cdot b⃗ = \frac{3 - 40}{3}$

Step 2.2.4

Subtract

40

$40$ from

3

$3$ .

a⃗ \cdot b⃗ = \frac{- 37}{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}

$a⃗ \cdot b⃗ = - \frac{37}{3}$

a⃗ \cdot b⃗ = - \frac{37}{3}

$a⃗ \cdot b⃗ = - \frac{37}{3}$

a⃗ \cdot b⃗ = - \frac{37}{3}

$a⃗ \cdot b⃗ = - \frac{37}{3}$

Step 3

Find the magnitude of

a⃗

$a⃗$ .

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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}}

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

Step 3.2

Simplify.

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Step 3.2.1

One to any power is one.

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

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

Step 3.2.2

Apply the product rule to

\frac{5}{3}

$\frac{5}{3}$ .

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

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

Step 3.2.3

Raise

5

$5$ to the power of

2

$2$ .

| a⃗ | = \sqrt{1 + \frac{25}{3^{2}}}

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

Step 3.2.4

Raise

3

$3$ to the power of

2

$2$ .

| a⃗ | = \sqrt{1 + \frac{25}{9}}

$| a⃗ | = \sqrt{1 + \frac{25}{9}}$

Step 3.2.5

Write

1

$1$ as a fraction with a common denominator.

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

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

Step 3.2.6

Combine the numerators over the common denominator.

| a⃗ | = \sqrt{\frac{9 + 25}{9}}

$| a⃗ | = \sqrt{\frac{9 + 25}{9}}$

Step 3.2.7

Add

9

$9$ and

25

$25$ .

| a⃗ | = \sqrt{\frac{34}{9}}

$| a⃗ | = \sqrt{\frac{34}{9}}$

Step 3.2.8

Rewrite

\sqrt{\frac{34}{9}}

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

\frac{\sqrt{34}}{\sqrt{9}}

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

| a⃗ | = \frac{\sqrt{34}}{\sqrt{9}}

$| a⃗ | = \frac{\sqrt{34}}{\sqrt{9}}$

Step 3.2.9

Simplify the denominator.

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Step 3.2.9.1

Rewrite

9

$9$ as

3^{2}

$3^{2}$ .

| a⃗ | = \frac{\sqrt{34}}{\sqrt{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}

$| a⃗ | = \frac{\sqrt{34}}{3}$

| a⃗ | = \frac{\sqrt{34}}{3}

$| a⃗ | = \frac{\sqrt{34}}{3}$

| a⃗ | = \frac{\sqrt{34}}{3}

$| a⃗ | = \frac{\sqrt{34}}{3}$

| a⃗ | = \frac{\sqrt{34}}{3}

$| a⃗ | = \frac{\sqrt{34}}{3}$

Step 4

Find the magnitude of

b⃗

$b⃗$ .

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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}}

$| b⃗ | = \sqrt{1^{2} + {(- 8)}^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

One to any power is one.

| b⃗ | = \sqrt{1 + {(- 8)}^{2}}

$| b⃗ | = \sqrt{1 + {(- 8)}^{2}}$

Step 4.2.2

Raise

- 8

$- 8$ to the power of

2

$2$ .

| b⃗ | = \sqrt{1 + 64}

$| b⃗ | = \sqrt{1 + 64}$

Step 4.2.3

Add

1

$1$ and

64

$64$ .

| b⃗ | = \sqrt{65}

$| b⃗ | = \sqrt{65}$

| b⃗ | = \sqrt{65}

$| b⃗ | = \sqrt{65}$

| b⃗ | = \sqrt{65}

$| b⃗ | = \sqrt{65}$

Step 5

Substitute the values into the formula.

θ = arccos ⎛ ⎜ ⎝ \frac{- \frac{37}{3}}{\frac{\sqrt{34}}{3} \sqrt{65}} ⎞ ⎟ ⎠

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

Step 6

Simplify.

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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}} ⎞ ⎟ ⎠

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

Step 6.2

Combine

\frac{\sqrt{34}}{3}

$\frac{\sqrt{34}}{3}$ and

\sqrt{65}

$\sqrt{65}$ .

θ = arccos ⎛ ⎜ ⎝ - \frac{37}{3} \cdot \frac{1}{\frac{\sqrt{34} \sqrt{65}}{3}} ⎞ ⎟ ⎠

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

Step 6.3

Simplify the numerator.

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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}} ⎞ ⎟ ⎠

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

Step 6.3.2

Multiply

34

$34$ by

65

$65$ .

θ = arccos ⎛ ⎜ ⎝ - \frac{37}{3} \cdot \frac{1}{\frac{\sqrt{2210}}{3}} ⎞ ⎟ ⎠

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

θ = arccos ⎛ ⎜ ⎝ - \frac{37}{3} \cdot \frac{1}{\frac{\sqrt{2210}}{3}} ⎞ ⎟ ⎠

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

Step 6.4

Multiply the numerator by the reciprocal of the denominator.

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

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

Step 6.5

Multiply

\frac{3}{\sqrt{2210}}

$\frac{3}{\sqrt{2210}}$ by

1

$1$ .

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

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

Step 6.6

Cancel the common factor of

3

$3$ .

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Step 6.6.1

Move the leading negative in

- \frac{37}{3}

$- \frac{37}{3}$ into the numerator.

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

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

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.

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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

$2210$ .

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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

$2$ .

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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

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

$θ = 141.91122711$