Trigonometry Examples

$(3, - 4)$ , $(- 5, 7)$

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.

$a⃗ \cdot b⃗ = 3 \cdot - 5 - 4 \cdot 7$

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 $3$ by $- 5$ .

$a⃗ \cdot b⃗ = - 15 - 4 \cdot 7$

Step 2.2.1.2

Multiply $- 4$ by $7$ .

$a⃗ \cdot b⃗ = - 15 - 28$

Step 2.2.2

Subtract $28$ from $- 15$ .

$a⃗ \cdot b⃗ = - 43$

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.

$| a⃗ | = \sqrt{3^{2} + {(- 4)}^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

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

$| a⃗ | = \sqrt{9 + {(- 4)}^{2}}$

Step 3.2.2

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

$| a⃗ | = \sqrt{9 + 16}$

Step 3.2.3

Add $9$ and $16$ .

$| a⃗ | = \sqrt{25}$

Step 3.2.4

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

$| a⃗ | = \sqrt{5^{2}}$

Step 3.2.5

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

$| a⃗ | = 5$

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.

$| b⃗ | = \sqrt{{(- 5)}^{2} + 7^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

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

$| b⃗ | = \sqrt{25 + 7^{2}}$

Step 4.2.2

Raise $7$ to the power of $2$ .

$| b⃗ | = \sqrt{25 + 49}$

Step 4.2.3

Add $25$ and $49$ .

$| b⃗ | = \sqrt{74}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{- 43}{5 \sqrt{74}})$

Step 6

Simplify.

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Step 6.1

Move the negative in front of the fraction.

$θ = \arccos (- \frac{43}{5 \sqrt{74}})$

Step 6.2

Multiply $\frac{43}{5 \sqrt{74}}$ by $\frac{\sqrt{74}}{\sqrt{74}}$ .

$θ = \arccos (- (\frac{43}{5 \sqrt{74}} \cdot \frac{\sqrt{74}}{\sqrt{74}}))$

Step 6.3

Combine and simplify the denominator.

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Step 6.3.1

Multiply $\frac{43}{5 \sqrt{74}}$ by $\frac{\sqrt{74}}{\sqrt{74}}$ .

$θ = \arccos (- \frac{43 \sqrt{74}}{5 \sqrt{74} \sqrt{74}})$

Step 6.3.2

Move $\sqrt{74}$ .

$θ = \arccos (- \frac{43 \sqrt{74}}{5 (\sqrt{74} \sqrt{74})})$

Step 6.3.3

Raise $\sqrt{74}$ to the power of $1$ .

$θ = \arccos (- \frac{43 \sqrt{74}}{5 ({\sqrt{74}}^{1} \sqrt{74})})$

Step 6.3.4

Raise $\sqrt{74}$ to the power of $1$ .

$θ = \arccos (- \frac{43 \sqrt{74}}{5 ({\sqrt{74}}^{1} {\sqrt{74}}^{1})})$

Step 6.3.5

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

$θ = \arccos (- \frac{43 \sqrt{74}}{5 {\sqrt{74}}^{1 + 1}})$

Step 6.3.6

Add $1$ and $1$ .

$θ = \arccos (- \frac{43 \sqrt{74}}{5 {\sqrt{74}}^{2}})$

Step 6.3.7

Rewrite ${\sqrt{74}}^{2}$ as $74$ .

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Step 6.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{74}$ as $74^{\frac{1}{2}}$ .

$θ = \arccos (- \frac{43 \sqrt{74}}{5 {(74^{\frac{1}{2}})}^{2}})$

Step 6.3.7.2

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

$θ = \arccos (- \frac{43 \sqrt{74}}{5 \cdot 74^{\frac{1}{2} \cdot 2}})$

Step 6.3.7.3

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

$θ = \arccos (- \frac{43 \sqrt{74}}{5 \cdot 74^{\frac{2}{2}}})$

Step 6.3.7.4

Cancel the common factor of $2$ .

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Step 6.3.7.4.1

Cancel the common factor.

$θ = \arccos (- \frac{43 \sqrt{74}}{5 \cdot 74^{\frac{2}{2}}})$

Step 6.3.7.4.2

Rewrite the expression.

$θ = \arccos (- \frac{43 \sqrt{74}}{5 \cdot 74^{1}})$

Step 6.3.7.5

Evaluate the exponent.

$θ = \arccos (- \frac{43 \sqrt{74}}{5 \cdot 74})$

Step 6.4

Multiply $5$ by $74$ .

$θ = \arccos (- \frac{43 \sqrt{74}}{370})$

Step 6.5

Evaluate $\arccos (- \frac{43 \sqrt{74}}{370})$ .

$θ = 178.66778014$