Trigonometry Examples

$(0, 9)$ , $(5, 80)$

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.

$a⃗ \cdot b⃗ = 0 \cdot 5 + 9 \cdot 80$

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 $0$ by $5$ .

$a⃗ \cdot b⃗ = 0 + 9 \cdot 80$

Step 2.2.1.2

Multiply $9$ by $80$ .

$a⃗ \cdot b⃗ = 0 + 720$

Step 2.2.2

Add $0$ and $720$ .

$a⃗ \cdot b⃗ = 720$

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.

$| a⃗ | = \sqrt{0^{2} + 9^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

Raising $0$ to any positive power yields $0$ .

$| a⃗ | = \sqrt{0 + 9^{2}}$

Step 3.2.2

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

$| a⃗ | = \sqrt{0 + 81}$

Step 3.2.3

Add $0$ and $81$ .

$| a⃗ | = \sqrt{81}$

Step 3.2.4

Rewrite $81$ as $9^{2}$ .

$| a⃗ | = \sqrt{9^{2}}$

Step 3.2.5

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

$| a⃗ | = 9$

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.

$| b⃗ | = \sqrt{5^{2} + 80^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

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

$| b⃗ | = \sqrt{25 + 80^{2}}$

Step 4.2.2

Raise $80$ to the power of $2$ .

$| b⃗ | = \sqrt{25 + 6400}$

Step 4.2.3

Add $25$ and $6400$ .

$| b⃗ | = \sqrt{6425}$

Step 4.2.4

Rewrite $6425$ as $5^{2} \cdot 257$ .

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Step 4.2.4.1

Factor $25$ out of $6425$ .

$| b⃗ | = \sqrt{25 (257)}$

Step 4.2.4.2

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

$| b⃗ | = \sqrt{5^{2} \cdot 257}$

Step 4.2.5

Pull terms out from under the radical.

$| b⃗ | = 5 \sqrt{257}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{720}{9 (5 \sqrt{257})})$

Step 6

Simplify.

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Step 6.1

Cancel the common factor of $720$ and $9$ .

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Step 6.1.1

Factor $9$ out of $720$ .

$θ = \arccos (\frac{9 \cdot 80}{9 (5 \sqrt{257})})$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Cancel the common factor.

$θ = \arccos (\frac{9 \cdot 80}{9 (5 \sqrt{257})})$

Step 6.1.2.2

Rewrite the expression.

$θ = \arccos (\frac{80}{5 \sqrt{257}})$

Step 6.2

Cancel the common factor of $80$ and $5$ .

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Step 6.2.1

Factor $5$ out of $80$ .

$θ = \arccos (\frac{5 \cdot 16}{5 \sqrt{257}})$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $5$ out of $5 \sqrt{257}$ .

$θ = \arccos (\frac{5 \cdot 16}{5 (\sqrt{257})})$

Step 6.2.2.2

Cancel the common factor.

$θ = \arccos (\frac{5 \cdot 16}{5 \sqrt{257}})$

Step 6.2.2.3

Rewrite the expression.

$θ = \arccos (\frac{16}{\sqrt{257}})$

Step 6.3

Multiply $\frac{16}{\sqrt{257}}$ by $\frac{\sqrt{257}}{\sqrt{257}}$ .

$θ = \arccos (\frac{16}{\sqrt{257}} \cdot \frac{\sqrt{257}}{\sqrt{257}})$

Step 6.4

Combine and simplify the denominator.

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Step 6.4.1

Multiply $\frac{16}{\sqrt{257}}$ by $\frac{\sqrt{257}}{\sqrt{257}}$ .

$θ = \arccos (\frac{16 \sqrt{257}}{\sqrt{257} \sqrt{257}})$

Step 6.4.2

Raise $\sqrt{257}$ to the power of $1$ .

$θ = \arccos (\frac{16 \sqrt{257}}{{\sqrt{257}}^{1} \sqrt{257}})$

Step 6.4.3

Raise $\sqrt{257}$ to the power of $1$ .

$θ = \arccos (\frac{16 \sqrt{257}}{{\sqrt{257}}^{1} {\sqrt{257}}^{1}})$

Step 6.4.4

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

$θ = \arccos (\frac{16 \sqrt{257}}{{\sqrt{257}}^{1 + 1}})$

Step 6.4.5

Add $1$ and $1$ .

$θ = \arccos (\frac{16 \sqrt{257}}{{\sqrt{257}}^{2}})$

Step 6.4.6

Rewrite ${\sqrt{257}}^{2}$ as $257$ .

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Step 6.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{257}$ as $257^{\frac{1}{2}}$ .

$θ = \arccos (\frac{16 \sqrt{257}}{{(257^{\frac{1}{2}})}^{2}})$

Step 6.4.6.2

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

$θ = \arccos (\frac{16 \sqrt{257}}{257^{\frac{1}{2} \cdot 2}})$

Step 6.4.6.3

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

$θ = \arccos (\frac{16 \sqrt{257}}{257^{\frac{2}{2}}})$

Step 6.4.6.4

Cancel the common factor of $2$ .

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Step 6.4.6.4.1

Cancel the common factor.

$θ = \arccos (\frac{16 \sqrt{257}}{257^{\frac{2}{2}}})$

Step 6.4.6.4.2

Rewrite the expression.

$θ = \arccos (\frac{16 \sqrt{257}}{257^{1}})$

Step 6.4.6.5

Evaluate the exponent.

$θ = \arccos (\frac{16 \sqrt{257}}{257})$

Step 6.5

Evaluate $\arccos (\frac{16 \sqrt{257}}{257})$ .

$θ = 3.57633437$