Trigonometry Examples

${(\tan (45 °) + \tan (60 °))}^{2}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

The exact value of $\tan (45 °)$ is $1$ .

${(1 + \tan (60 °))}^{2}$

Step 1.1.2

The exact value of $\tan (60 °)$ is $\sqrt{3}$ .

${(1 + \sqrt{3})}^{2}$

Step 1.2

Rewrite ${(1 + \sqrt{3})}^{2}$ as $(1 + \sqrt{3}) (1 + \sqrt{3})$ .

$(1 + \sqrt{3}) (1 + \sqrt{3})$

Step 2

Expand $(1 + \sqrt{3}) (1 + \sqrt{3})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$1 (1 + \sqrt{3}) + \sqrt{3} (1 + \sqrt{3})$

Step 2.2

Apply the distributive property.

$1 \cdot 1 + 1 \sqrt{3} + \sqrt{3} (1 + \sqrt{3})$

Step 2.3

Apply the distributive property.

$1 \cdot 1 + 1 \sqrt{3} + \sqrt{3} \cdot 1 + \sqrt{3} \sqrt{3}$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $1$ by $1$ .

$1 + 1 \sqrt{3} + \sqrt{3} \cdot 1 + \sqrt{3} \sqrt{3}$

Step 3.1.2

Multiply $\sqrt{3}$ by $1$ .

$1 + \sqrt{3} + \sqrt{3} \cdot 1 + \sqrt{3} \sqrt{3}$

Step 3.1.3

Multiply $\sqrt{3}$ by $1$ .

$1 + \sqrt{3} + \sqrt{3} + \sqrt{3} \sqrt{3}$

Step 3.1.4

Combine using the product rule for radicals.

$1 + \sqrt{3} + \sqrt{3} + \sqrt{3 \cdot 3}$

Step 3.1.5

Multiply $3$ by $3$ .

$1 + \sqrt{3} + \sqrt{3} + \sqrt{9}$

Step 3.1.6

Rewrite $9$ as $3^{2}$ .

$1 + \sqrt{3} + \sqrt{3} + \sqrt{3^{2}}$

Step 3.1.7

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

$1 + \sqrt{3} + \sqrt{3} + 3$

Step 3.2

Add $1$ and $3$ .

$4 + \sqrt{3} + \sqrt{3}$

Step 3.3

Add $\sqrt{3}$ and $\sqrt{3}$ .

$4 + 2 \sqrt{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$4 + 2 \sqrt{3}$

Decimal Form:

$7.46410161 \dots$