Trigonometry Examples

$a = (5, 5)$ , $b = (- 3, - 2)$

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.

$a⃗ \cdot b⃗ = 5 \cdot - 3 + 5 \cdot - 2$

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 $5$ by $- 3$ .

$a⃗ \cdot b⃗ = - 15 + 5 \cdot - 2$

Step 2.2.1.2

Multiply $5$ by $- 2$ .

$a⃗ \cdot b⃗ = - 15 - 10$

Step 2.2.2

Subtract $10$ from $- 15$ .

$a⃗ \cdot b⃗ = - 25$

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.

$| a⃗ | = \sqrt{5^{2} + 5^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

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

$| a⃗ | = \sqrt{25 + 5^{2}}$

Step 3.2.2

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

$| a⃗ | = \sqrt{25 + 25}$

Step 3.2.3

Add $25$ and $25$ .

$| a⃗ | = \sqrt{50}$

Step 3.2.4

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 3.2.4.1

Factor $25$ out of $50$ .

$| a⃗ | = \sqrt{25 (2)}$

Step 3.2.4.2

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

$| a⃗ | = \sqrt{5^{2} \cdot 2}$

Step 3.2.5

Pull terms out from under the radical.

$| a⃗ | = 5 \sqrt{2}$

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.

$| b⃗ | = \sqrt{{(- 3)}^{2} + {(- 2)}^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

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

$| b⃗ | = \sqrt{9 + {(- 2)}^{2}}$

Step 4.2.2

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

$| b⃗ | = \sqrt{9 + 4}$

Step 4.2.3

Add $9$ and $4$ .

$| b⃗ | = \sqrt{13}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{- 25}{5 \sqrt{2} \sqrt{13}})$

Step 6

Simplify.

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Step 6.1

Cancel the common factor of $- 25$ and $5$ .

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Step 6.1.1

Factor $5$ out of $- 25$ .

$θ = \arccos (\frac{5 \cdot - 5}{5 \sqrt{2} \sqrt{13}})$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $5$ out of $5 \sqrt{2} \sqrt{13}$ .

$θ = \arccos (\frac{5 \cdot - 5}{5 (\sqrt{2} \sqrt{13})})$

Step 6.1.2.2

Cancel the common factor.

$θ = \arccos (\frac{5 \cdot - 5}{5 (\sqrt{2} \sqrt{13})})$

Step 6.1.2.3

Rewrite the expression.

$θ = \arccos (\frac{- 5}{\sqrt{2} \sqrt{13}})$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Combine using the product rule for radicals.

$θ = \arccos (\frac{- 5}{\sqrt{2 \cdot 13}})$

Step 6.2.2

Multiply $2$ by $13$ .

$θ = \arccos (\frac{- 5}{\sqrt{26}})$

Step 6.3

Move the negative in front of the fraction.

$θ = \arccos (- \frac{5}{\sqrt{26}})$

Step 6.4

Multiply $\frac{5}{\sqrt{26}}$ by $\frac{\sqrt{26}}{\sqrt{26}}$ .

$θ = \arccos (- (\frac{5}{\sqrt{26}} \cdot \frac{\sqrt{26}}{\sqrt{26}}))$

Step 6.5

Combine and simplify the denominator.

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Step 6.5.1

Multiply $\frac{5}{\sqrt{26}}$ by $\frac{\sqrt{26}}{\sqrt{26}}$ .

$θ = \arccos (- \frac{5 \sqrt{26}}{\sqrt{26} \sqrt{26}})$

Step 6.5.2

Raise $\sqrt{26}$ to the power of $1$ .

$θ = \arccos (- \frac{5 \sqrt{26}}{{\sqrt{26}}^{1} \sqrt{26}})$

Step 6.5.3

Raise $\sqrt{26}$ to the power of $1$ .

$θ = \arccos (- \frac{5 \sqrt{26}}{{\sqrt{26}}^{1} {\sqrt{26}}^{1}})$

Step 6.5.4

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

$θ = \arccos (- \frac{5 \sqrt{26}}{{\sqrt{26}}^{1 + 1}})$

Step 6.5.5

Add $1$ and $1$ .

$θ = \arccos (- \frac{5 \sqrt{26}}{{\sqrt{26}}^{2}})$

Step 6.5.6

Rewrite ${\sqrt{26}}^{2}$ as $26$ .

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Step 6.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{26}$ as $26^{\frac{1}{2}}$ .

$θ = \arccos (- \frac{5 \sqrt{26}}{{(26^{\frac{1}{2}})}^{2}})$

Step 6.5.6.2

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

$θ = \arccos (- \frac{5 \sqrt{26}}{26^{\frac{1}{2} \cdot 2}})$

Step 6.5.6.3

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

$θ = \arccos (- \frac{5 \sqrt{26}}{26^{\frac{2}{2}}})$

Step 6.5.6.4

Cancel the common factor of $2$ .

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Step 6.5.6.4.1

Cancel the common factor.

$θ = \arccos (- \frac{5 \sqrt{26}}{26^{\frac{2}{2}}})$

Step 6.5.6.4.2

Rewrite the expression.

$θ = \arccos (- \frac{5 \sqrt{26}}{26^{1}})$

Step 6.5.6.5

Evaluate the exponent.

$θ = \arccos (- \frac{5 \sqrt{26}}{26})$

Step 6.6

Evaluate $\arccos (- \frac{5 \sqrt{26}}{26})$ .

$θ = 168.69006752$