Trigonometry Examples

$a = (- 8, 8)$ , $b = (- 6, 2)$

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.

$a⃗ \cdot b⃗ = - 8 \cdot - 6 + 8 \cdot 2$

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 $- 8$ by $- 6$ .

$a⃗ \cdot b⃗ = 48 + 8 \cdot 2$

Step 2.2.1.2

Multiply $8$ by $2$ .

$a⃗ \cdot b⃗ = 48 + 16$

Step 2.2.2

Add $48$ and $16$ .

$a⃗ \cdot b⃗ = 64$

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.

$| a⃗ | = \sqrt{{(- 8)}^{2} + 8^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

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

$| a⃗ | = \sqrt{64 + 8^{2}}$

Step 3.2.2

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

$| a⃗ | = \sqrt{64 + 64}$

Step 3.2.3

Add $64$ and $64$ .

$| a⃗ | = \sqrt{128}$

Step 3.2.4

Rewrite $128$ as $8^{2} \cdot 2$ .

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Step 3.2.4.1

Factor $64$ out of $128$ .

$| a⃗ | = \sqrt{64 (2)}$

Step 3.2.4.2

Rewrite $64$ as $8^{2}$ .

$| a⃗ | = \sqrt{8^{2} \cdot 2}$

Step 3.2.5

Pull terms out from under the radical.

$| a⃗ | = 8 \sqrt{2}$

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.

$| b⃗ | = \sqrt{{(- 6)}^{2} + 2^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

Raise $- 6$ to the power of $2$ .

$| b⃗ | = \sqrt{36 + 2^{2}}$

Step 4.2.2

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

$| b⃗ | = \sqrt{36 + 4}$

Step 4.2.3

Add $36$ and $4$ .

$| b⃗ | = \sqrt{40}$

Step 4.2.4

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 4.2.4.1

Factor $4$ out of $40$ .

$| b⃗ | = \sqrt{4 (10)}$

Step 4.2.4.2

Rewrite $4$ as $2^{2}$ .

$| b⃗ | = \sqrt{2^{2} \cdot 10}$

Step 4.2.5

Pull terms out from under the radical.

$| b⃗ | = 2 \sqrt{10}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{64}{8 \sqrt{2} (2 \sqrt{10})})$

Step 6

Simplify.

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Step 6.1

Cancel the common factor of $64$ and $8$ .

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Step 6.1.1

Factor $8$ out of $64$ .

$θ = \arccos (\frac{8 \cdot 8}{8 \sqrt{2} (2 \sqrt{10})})$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $8$ out of $8 \sqrt{2} (2 \sqrt{10})$ .

$θ = \arccos (\frac{8 \cdot 8}{8 (\sqrt{2} (2 \sqrt{10}))})$

Step 6.1.2.2

Cancel the common factor.

$θ = \arccos (\frac{8 \cdot 8}{8 (\sqrt{2} (2 \sqrt{10}))})$

Step 6.1.2.3

Rewrite the expression.

$θ = \arccos (\frac{8}{\sqrt{2} (2 \sqrt{10})})$

Step 6.2

Cancel the common factor of $8$ and $2$ .

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Step 6.2.1

Factor $2$ out of $8$ .

$θ = \arccos (\frac{2 \cdot 4}{\sqrt{2} (2 \sqrt{10})})$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $2$ out of $\sqrt{2} (2 \sqrt{10})$ .

$θ = \arccos (\frac{2 \cdot 4}{2 (\sqrt{2} (\sqrt{10}))})$

Step 6.2.2.2

Cancel the common factor.

$θ = \arccos (\frac{2 \cdot 4}{2 (\sqrt{2} (\sqrt{10}))})$

Step 6.2.2.3

Rewrite the expression.

$θ = \arccos (\frac{4}{\sqrt{2} (\sqrt{10})})$

Step 6.3

Simplify the denominator.

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Step 6.3.1

Combine using the product rule for radicals.

$θ = \arccos (\frac{4}{\sqrt{2 \cdot 10}})$

Step 6.3.2

Multiply $2$ by $10$ .

$θ = \arccos (\frac{4}{\sqrt{20}})$

Step 6.4

Simplify the denominator.

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Step 6.4.1

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 6.4.1.1

Factor $4$ out of $20$ .

$θ = \arccos (\frac{4}{\sqrt{4 (5)}})$

Step 6.4.1.2

Rewrite $4$ as $2^{2}$ .

$θ = \arccos (\frac{4}{\sqrt{2^{2} \cdot 5}})$

Step 6.4.2

Pull terms out from under the radical.

$θ = \arccos (\frac{4}{2 \sqrt{5}})$

Step 6.5

Cancel the common factor of $4$ and $2$ .

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Step 6.5.1

Factor $2$ out of $4$ .

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

Step 6.5.2

Cancel the common factors.

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Step 6.5.2.1

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

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

Step 6.5.2.2

Cancel the common factor.

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

Step 6.5.2.3

Rewrite the expression.

$θ = \arccos (\frac{2}{\sqrt{5}})$

Step 6.6

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 6.7

Combine and simplify the denominator.

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Step 6.7.1

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 6.7.2

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 6.7.3

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 6.7.4

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

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

Step 6.7.5

Add $1$ and $1$ .

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

Step 6.7.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 6.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

Step 6.7.6.2

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

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

Step 6.7.6.3

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

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

Step 6.7.6.4

Cancel the common factor of $2$ .

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Step 6.7.6.4.1

Cancel the common factor.

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

Step 6.7.6.4.2

Rewrite the expression.

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

Step 6.7.6.5

Evaluate the exponent.

$θ = \arccos (\frac{2 \sqrt{5}}{5})$

Step 6.8

Evaluate $\arccos (\frac{2 \sqrt{5}}{5})$ .

$θ = 26.56505117$