Trigonometry Examples

$(1.8, - 2)$ , $(3.8, - 1.4)$

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.

$a⃗ \cdot b⃗ = 1.8 \cdot 3.8 - 2 \cdot - 1.4$

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 $1.8$ by $3.8$ .

$a⃗ \cdot b⃗ = 6.84 - 2 \cdot - 1.4$

Step 2.2.1.2

Multiply $- 2$ by $- 1.4$ .

$a⃗ \cdot b⃗ = 6.84 + 2.8$

Step 2.2.2

Add $6.84$ and $2.8$ .

$a⃗ \cdot b⃗ = 9.64$

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.

$| a⃗ | = \sqrt{{1.8}^{2} + {(- 2)}^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

Raise $1.8$ to the power of $2$ .

$| a⃗ | = \sqrt{3.24 + {(- 2)}^{2}}$

Step 3.2.2

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

$| a⃗ | = \sqrt{3.24 + 4}$

Step 3.2.3

Add $3.24$ and $4$ .

$| a⃗ | = \sqrt{7.24}$

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.

$| b⃗ | = \sqrt{{3.8}^{2} + {(- 1.4)}^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

Raise $3.8$ to the power of $2$ .

$| b⃗ | = \sqrt{14.44 + {(- 1.4)}^{2}}$

Step 4.2.2

Raise $- 1.4$ to the power of $2$ .

$| b⃗ | = \sqrt{14.44 + 1.96}$

Step 4.2.3

Add $14.44$ and $1.96$ .

$| b⃗ | = \sqrt{16.4}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{9.64}{\sqrt{7.24} \sqrt{16.4}})$

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.

$θ = \arccos (\frac{9.64}{\sqrt{7.24 \cdot 16.4}})$

Step 6.1.2

Multiply $7.24$ by $16.4$ .

$θ = \arccos (\frac{9.64}{\sqrt{118.736}})$

Step 6.2

Multiply $\frac{9.64}{\sqrt{118.736}}$ by $\frac{\sqrt{118.736}}{\sqrt{118.736}}$ .

$θ = \arccos (\frac{9.64}{\sqrt{118.736}} \cdot \frac{\sqrt{118.736}}{\sqrt{118.736}})$

Step 6.3

Combine and simplify the denominator.

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Step 6.3.1

Multiply $\frac{9.64}{\sqrt{118.736}}$ by $\frac{\sqrt{118.736}}{\sqrt{118.736}}$ .

$θ = \arccos (\frac{9.64 \sqrt{118.736}}{\sqrt{118.736} \sqrt{118.736}})$

Step 6.3.2

Raise $\sqrt{118.736}$ to the power of $1$ .

$θ = \arccos (\frac{9.64 \sqrt{118.736}}{{\sqrt{118.736}}^{1} \sqrt{118.736}})$

Step 6.3.3

Raise $\sqrt{118.736}$ to the power of $1$ .

$θ = \arccos (\frac{9.64 \sqrt{118.736}}{{\sqrt{118.736}}^{1} {\sqrt{118.736}}^{1}})$

Step 6.3.4

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

$θ = \arccos (\frac{9.64 \sqrt{118.736}}{{\sqrt{118.736}}^{1 + 1}})$

Step 6.3.5

Add $1$ and $1$ .

$θ = \arccos (\frac{9.64 \sqrt{118.736}}{{\sqrt{118.736}}^{2}})$

Step 6.3.6

Rewrite ${\sqrt{118.736}}^{2}$ as $118.736$ .

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Step 6.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{118.736}$ as ${118.736}^{\frac{1}{2}}$ .

$θ = \arccos (\frac{9.64 \sqrt{118.736}}{{({118.736}^{\frac{1}{2}})}^{2}})$

Step 6.3.6.2

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

$θ = \arccos (\frac{9.64 \sqrt{118.736}}{{118.736}^{\frac{1}{2} \cdot 2}})$

Step 6.3.6.3

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

$θ = \arccos (\frac{9.64 \sqrt{118.736}}{{118.736}^{\frac{2}{2}}})$

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.

$θ = \arccos (\frac{9.64 \sqrt{118.736}}{{118.736}^{\frac{2}{2}}})$

Step 6.3.6.4.2

Rewrite the expression.

$θ = \arccos (\frac{9.64 \sqrt{118.736}}{{118.736}^{1}})$

Step 6.3.6.5

Evaluate the exponent.

$θ = \arccos (\frac{9.64 \sqrt{118.736}}{118.736})$

Step 6.4

Multiply $9.64$ by $\sqrt{118.736}$ .

$θ = \arccos (\frac{105.04327196}{118.736})$

Step 6.5

Divide $105.04327196$ by $118.736$ .

$θ = \arccos (0.88467922)$

Step 6.6

Evaluate $\arccos (0.88467922)$ .

$θ = 27.78792807$