Trigonometry Examples

$(14, - 3)$ , $(1, 4)$

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.

$a⃗ \cdot b⃗ = 14 \cdot 1 - 3 \cdot 4$

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 $14$ by $1$ .

$a⃗ \cdot b⃗ = 14 - 3 \cdot 4$

Step 2.2.1.2

Multiply $- 3$ by $4$ .

$a⃗ \cdot b⃗ = 14 - 12$

Step 2.2.2

Subtract $12$ from $14$ .

$a⃗ \cdot b⃗ = 2$

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.

$| a⃗ | = \sqrt{14^{2} + {(- 3)}^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

Raise $14$ to the power of $2$ .

$| a⃗ | = \sqrt{196 + {(- 3)}^{2}}$

Step 3.2.2

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

$| a⃗ | = \sqrt{196 + 9}$

Step 3.2.3

Add $196$ and $9$ .

$| a⃗ | = \sqrt{205}$

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.

$| b⃗ | = \sqrt{1^{2} + 4^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

One to any power is one.

$| b⃗ | = \sqrt{1 + 4^{2}}$

Step 4.2.2

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

$| b⃗ | = \sqrt{1 + 16}$

Step 4.2.3

Add $1$ and $16$ .

$| b⃗ | = \sqrt{17}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{2}{\sqrt{205} \sqrt{17}})$

Step 6

Simplify.

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Step 6.1

Simplify the denominator.

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Step 6.1.1

Combine using the product rule for radicals.

$θ = \arccos (\frac{2}{\sqrt{205 \cdot 17}})$

Step 6.1.2

Multiply $205$ by $17$ .

$θ = \arccos (\frac{2}{\sqrt{3485}})$

Step 6.2

Multiply $\frac{2}{\sqrt{3485}}$ by $\frac{\sqrt{3485}}{\sqrt{3485}}$ .

$θ = \arccos (\frac{2}{\sqrt{3485}} \cdot \frac{\sqrt{3485}}{\sqrt{3485}})$

Step 6.3

Combine and simplify the denominator.

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Step 6.3.1

Multiply $\frac{2}{\sqrt{3485}}$ by $\frac{\sqrt{3485}}{\sqrt{3485}}$ .

$θ = \arccos (\frac{2 \sqrt{3485}}{\sqrt{3485} \sqrt{3485}})$

Step 6.3.2

Raise $\sqrt{3485}$ to the power of $1$ .

$θ = \arccos (\frac{2 \sqrt{3485}}{{\sqrt{3485}}^{1} \sqrt{3485}})$

Step 6.3.3

Raise $\sqrt{3485}$ to the power of $1$ .

$θ = \arccos (\frac{2 \sqrt{3485}}{{\sqrt{3485}}^{1} {\sqrt{3485}}^{1}})$

Step 6.3.4

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

$θ = \arccos (\frac{2 \sqrt{3485}}{{\sqrt{3485}}^{1 + 1}})$

Step 6.3.5

Add $1$ and $1$ .

$θ = \arccos (\frac{2 \sqrt{3485}}{{\sqrt{3485}}^{2}})$

Step 6.3.6

Rewrite ${\sqrt{3485}}^{2}$ as $3485$ .

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Step 6.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3485}$ as $3485^{\frac{1}{2}}$ .

$θ = \arccos (\frac{2 \sqrt{3485}}{{(3485^{\frac{1}{2}})}^{2}})$

Step 6.3.6.2

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

$θ = \arccos (\frac{2 \sqrt{3485}}{3485^{\frac{1}{2} \cdot 2}})$

Step 6.3.6.3

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

$θ = \arccos (\frac{2 \sqrt{3485}}{3485^{\frac{2}{2}}})$

Step 6.3.6.4

Cancel the common factor of $2$ .

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Step 6.3.6.4.1

Cancel the common factor.

$θ = \arccos (\frac{2 \sqrt{3485}}{3485^{\frac{2}{2}}})$

Step 6.3.6.4.2

Rewrite the expression.

$θ = \arccos (\frac{2 \sqrt{3485}}{3485^{1}})$

Step 6.3.6.5

Evaluate the exponent.

$θ = \arccos (\frac{2 \sqrt{3485}}{3485})$

Step 6.4

Evaluate $\arccos (\frac{2 \sqrt{3485}}{3485})$ .

$θ = 88.0585136$