Trigonometry Examples

$(4, - 3)$ , $(9, 4)$

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.

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Step 2.1

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

$a⃗ \cdot b⃗ = 4 \cdot 9 - 3 \cdot 4$

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 $4$ by $9$ .

$a⃗ \cdot b⃗ = 36 - 3 \cdot 4$

Step 2.2.1.2

Multiply $- 3$ by $4$ .

$a⃗ \cdot b⃗ = 36 - 12$

Step 2.2.2

Subtract $12$ from $36$ .

$a⃗ \cdot b⃗ = 24$

Step 3

Find the magnitude of $a⃗$ .

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Step 3.1

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

$| a⃗ | = \sqrt{4^{2} + {(- 3)}^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

Raise $4$ to the power of $2$ .

$| a⃗ | = \sqrt{16 + {(- 3)}^{2}}$

Step 3.2.2

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

$| a⃗ | = \sqrt{16 + 9}$

Step 3.2.3

Add $16$ and $9$ .

$| a⃗ | = \sqrt{25}$

Step 3.2.4

Rewrite $25$ as $5^{2}$ .

$| a⃗ | = \sqrt{5^{2}}$

Step 3.2.5

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

$| a⃗ | = 5$

Step 4

Find the magnitude of $b⃗$ .

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Step 4.1

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

$| b⃗ | = \sqrt{9^{2} + 4^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

Raise $9$ to the power of $2$ .

$| b⃗ | = \sqrt{81 + 4^{2}}$

Step 4.2.2

Raise $4$ to the power of $2$ .

$| b⃗ | = \sqrt{81 + 16}$

Step 4.2.3

Add $81$ and $16$ .

$| b⃗ | = \sqrt{97}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{24}{5 \sqrt{97}})$

Step 6

Simplify.

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Step 6.1

Multiply $\frac{24}{5 \sqrt{97}}$ by $\frac{\sqrt{97}}{\sqrt{97}}$ .

$θ = \arccos (\frac{24}{5 \sqrt{97}} \cdot \frac{\sqrt{97}}{\sqrt{97}})$

Step 6.2

Combine and simplify the denominator.

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Step 6.2.1

Multiply $\frac{24}{5 \sqrt{97}}$ by $\frac{\sqrt{97}}{\sqrt{97}}$ .

$θ = \arccos (\frac{24 \sqrt{97}}{5 \sqrt{97} \sqrt{97}})$

Step 6.2.2

Move $\sqrt{97}$ .

$θ = \arccos (\frac{24 \sqrt{97}}{5 (\sqrt{97} \sqrt{97})})$

Step 6.2.3

Raise $\sqrt{97}$ to the power of $1$ .

$θ = \arccos (\frac{24 \sqrt{97}}{5 ({\sqrt{97}}^{1} \sqrt{97})})$

Step 6.2.4

Raise $\sqrt{97}$ to the power of $1$ .

$θ = \arccos (\frac{24 \sqrt{97}}{5 ({\sqrt{97}}^{1} {\sqrt{97}}^{1})})$

Step 6.2.5

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

$θ = \arccos (\frac{24 \sqrt{97}}{5 {\sqrt{97}}^{1 + 1}})$

Step 6.2.6

Add $1$ and $1$ .

$θ = \arccos (\frac{24 \sqrt{97}}{5 {\sqrt{97}}^{2}})$

Step 6.2.7

Rewrite ${\sqrt{97}}^{2}$ as $97$ .

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Step 6.2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{97}$ as $97^{\frac{1}{2}}$ .

$θ = \arccos (\frac{24 \sqrt{97}}{5 {(97^{\frac{1}{2}})}^{2}})$

Step 6.2.7.2

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

$θ = \arccos (\frac{24 \sqrt{97}}{5 \cdot 97^{\frac{1}{2} \cdot 2}})$

Step 6.2.7.3

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

$θ = \arccos (\frac{24 \sqrt{97}}{5 \cdot 97^{\frac{2}{2}}})$

Step 6.2.7.4

Cancel the common factor of $2$ .

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Step 6.2.7.4.1

Cancel the common factor.

$θ = \arccos (\frac{24 \sqrt{97}}{5 \cdot 97^{\frac{2}{2}}})$

Step 6.2.7.4.2

Rewrite the expression.

$θ = \arccos (\frac{24 \sqrt{97}}{5 \cdot 97^{1}})$

Step 6.2.7.5

Evaluate the exponent.

$θ = \arccos (\frac{24 \sqrt{97}}{5 \cdot 97})$

Step 6.3

Multiply $5$ by $97$ .

$θ = \arccos (\frac{24 \sqrt{97}}{485})$

Step 6.4

Evaluate $\arccos (\frac{24 \sqrt{97}}{485})$ .

$θ = 60.83238662$