Trigonometry Examples

$(6, 10)$ ,

$(12, 5)$

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.

$a⃗ \cdot b⃗ = 6 \cdot 12 + 10 \cdot 5$

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

$6$ by

$12$ .

$a⃗ \cdot b⃗ = 72 + 10 \cdot 5$

Step 2.2.1.2

Multiply

$10$ by

$5$ .

$a⃗ \cdot b⃗ = 72 + 50$

Step 2.2.2

Add

$72$ and

$50$ .

$a⃗ \cdot b⃗ = 122$

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.

$| a⃗ | = \sqrt{6^{2} + 10^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

Raise

$6$ to the power of

$2$ .

$| a⃗ | = \sqrt{36 + 10^{2}}$

Step 3.2.2

Raise

$10$ to the power of

$2$ .

$| a⃗ | = \sqrt{36 + 100}$

Step 3.2.3

Add

$36$ and

$100$ .

$| a⃗ | = \sqrt{136}$

Step 3.2.4

Rewrite

$136$ as

$2^{2} \cdot 34$ .

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Step 3.2.4.1

Factor

$4$ out of

$136$ .

$| a⃗ | = \sqrt{4 (34)}$

Step 3.2.4.2

Rewrite

$4$ as

$2^{2}$ .

$| a⃗ | = \sqrt{2^{2} \cdot 34}$

Step 3.2.5

Pull terms out from under the radical.

$| a⃗ | = 2 \sqrt{34}$

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.

$| b⃗ | = \sqrt{12^{2} + 5^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

Raise

$12$ to the power of

$2$ .

$| b⃗ | = \sqrt{144 + 5^{2}}$

Step 4.2.2

Raise

$5$ to the power of

$2$ .

$| b⃗ | = \sqrt{144 + 25}$

Step 4.2.3

Add

$144$ and

$25$ .

$| b⃗ | = \sqrt{169}$

Step 4.2.4

Rewrite

$169$ as

$13^{2}$ .

$| b⃗ | = \sqrt{13^{2}}$

Step 4.2.5

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

$| b⃗ | = 13$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{122}{2 \sqrt{34} \cdot 13})$

Step 6

Simplify.

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Step 6.1

Cancel the common factor of

$122$ and

$2$ .

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Step 6.1.1

Factor

$2$ out of

$122$ .

$θ = \arccos (\frac{2 \cdot 61}{2 \sqrt{34} \cdot 13})$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor

$2$ out of

$2 \sqrt{34} \cdot 13$ .

$θ = \arccos (\frac{2 \cdot 61}{2 (\sqrt{34} \cdot 13)})$

Step 6.1.2.2

Cancel the common factor.

$θ = \arccos (\frac{2 \cdot 61}{2 (\sqrt{34} \cdot 13)})$

Step 6.1.2.3

Rewrite the expression.

$θ = \arccos (\frac{61}{\sqrt{34} \cdot 13})$

Step 6.2

Move

$13$ to the left of

$\sqrt{34}$ .

$θ = \arccos (\frac{61}{13 \sqrt{34}})$

Step 6.3

Multiply

$\frac{61}{13 \sqrt{34}}$ by

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

$θ = \arccos (\frac{61}{13 \sqrt{34}} \cdot \frac{\sqrt{34}}{\sqrt{34}})$

Step 6.4

Combine and simplify the denominator.

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Step 6.4.1

Multiply

$\frac{61}{13 \sqrt{34}}$ by

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

$θ = \arccos (\frac{61 \sqrt{34}}{13 \sqrt{34} \sqrt{34}})$

Step 6.4.2

Move

$\sqrt{34}$ .

$θ = \arccos (\frac{61 \sqrt{34}}{13 (\sqrt{34} \sqrt{34})})$

Step 6.4.3

Raise

$\sqrt{34}$ to the power of

$1$ .

$θ = \arccos (\frac{61 \sqrt{34}}{13 ({\sqrt{34}}^{1} \sqrt{34})})$

Step 6.4.4

Raise

$\sqrt{34}$ to the power of

$1$ .

$θ = \arccos (\frac{61 \sqrt{34}}{13 ({\sqrt{34}}^{1} {\sqrt{34}}^{1})})$

Step 6.4.5

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$θ = \arccos (\frac{61 \sqrt{34}}{13 {\sqrt{34}}^{1 + 1}})$

Step 6.4.6

Add

$1$ and

$1$ .

$θ = \arccos (\frac{61 \sqrt{34}}{13 {\sqrt{34}}^{2}})$

Step 6.4.7

Rewrite

${\sqrt{34}}^{2}$ as

$34$ .

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Step 6.4.7.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{34}$ as

$34^{\frac{1}{2}}$ .

$θ = \arccos (\frac{61 \sqrt{34}}{13 {(34^{\frac{1}{2}})}^{2}})$

Step 6.4.7.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$θ = \arccos (\frac{61 \sqrt{34}}{13 \cdot 34^{\frac{1}{2} \cdot 2}})$

Step 6.4.7.3

Combine

$\frac{1}{2}$ and

$2$ .

$θ = \arccos (\frac{61 \sqrt{34}}{13 \cdot 34^{\frac{2}{2}}})$

Step 6.4.7.4

Cancel the common factor of

$2$ .

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Step 6.4.7.4.1

Cancel the common factor.

$θ = \arccos (\frac{61 \sqrt{34}}{13 \cdot 34^{\frac{2}{2}}})$

Step 6.4.7.4.2

Rewrite the expression.

$θ = \arccos (\frac{61 \sqrt{34}}{13 \cdot 34^{1}})$

Step 6.4.7.5

Evaluate the exponent.

$θ = \arccos (\frac{61 \sqrt{34}}{13 \cdot 34})$

Step 6.5

Multiply

$13$ by

$34$ .

$θ = \arccos (\frac{61 \sqrt{34}}{442})$

Step 6.6

Evaluate

$\arccos (\frac{61 \sqrt{34}}{442})$ .

$θ = 36.41637851$