Precalculus Examples

$2 \tan^{2} (300) + 3 \sin^{2} (150) - \cos^{2} (180)$

Step 1

Simplify each term.

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Step 1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the fourth quadrant.

$2 {(- \tan (60))}^{2} + 3 \sin^{2} (150) - \cos^{2} (180)$

Step 1.2

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

$2 {(- \sqrt{3})}^{2} + 3 \sin^{2} (150) - \cos^{2} (180)$

Step 1.3

Apply the product rule to $- \sqrt{3}$ .

$2 ({(- 1)}^{2} {\sqrt{3}}^{2}) + 3 \sin^{2} (150) - \cos^{2} (180)$

Step 1.4

Raise $- 1$ to the power of $2$ .

$2 (1 {\sqrt{3}}^{2}) + 3 \sin^{2} (150) - \cos^{2} (180)$

Step 1.5

Multiply ${\sqrt{3}}^{2}$ by $1$ .

$2 {\sqrt{3}}^{2} + 3 \sin^{2} (150) - \cos^{2} (180)$

Step 1.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$2 {(3^{\frac{1}{2}})}^{2} + 3 \sin^{2} (150) - \cos^{2} (180)$

Step 1.6.2

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

$2 \cdot 3^{\frac{1}{2} \cdot 2} + 3 \sin^{2} (150) - \cos^{2} (180)$

Step 1.6.3

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

$2 \cdot 3^{\frac{2}{2}} + 3 \sin^{2} (150) - \cos^{2} (180)$

Step 1.6.4

Cancel the common factor of $2$ .

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Step 1.6.4.1

Cancel the common factor.

$2 \cdot 3^{\frac{2}{2}} + 3 \sin^{2} (150) - \cos^{2} (180)$

Step 1.6.4.2

Rewrite the expression.

$2 \cdot 3^{1} + 3 \sin^{2} (150) - \cos^{2} (180)$

Step 1.6.5

Evaluate the exponent.

$2 \cdot 3 + 3 \sin^{2} (150) - \cos^{2} (180)$

Step 1.7

Multiply $2$ by $3$ .

$6 + 3 \sin^{2} (150) - \cos^{2} (180)$

Step 1.8

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$6 + 3 \sin^{2} (30) - \cos^{2} (180)$

Step 1.9

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$6 + 3 {(\frac{1}{2})}^{2} - \cos^{2} (180)$

Step 1.10

Apply the product rule to $\frac{1}{2}$ .

$6 + 3 \frac{1^{2}}{2^{2}} - \cos^{2} (180)$

Step 1.11

One to any power is one.

$6 + 3 \frac{1}{2^{2}} - \cos^{2} (180)$

Step 1.12

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

$6 + 3 (\frac{1}{4}) - \cos^{2} (180)$

Step 1.13

Combine $3$ and $\frac{1}{4}$ .

$6 + \frac{3}{4} - \cos^{2} (180)$

Step 1.14

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$6 + \frac{3}{4} - {(- \cos (0))}^{2}$

Step 1.15

The exact value of $\cos (0)$ is $1$ .

$6 + \frac{3}{4} - {(- 1 \cdot 1)}^{2}$

Step 1.16

Multiply $- 1$ by $1$ .

$6 + \frac{3}{4} - {(- 1)}^{2}$

Step 1.17

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 1.17.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 1.17.1.1

Raise $- 1$ to the power of $1$ .

$6 + \frac{3}{4} + {(- 1)}^{1} {(- 1)}^{2}$

Step 1.17.1.2

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

$6 + \frac{3}{4} + {(- 1)}^{1 + 2}$

Step 1.17.2

Add $1$ and $2$ .

$6 + \frac{3}{4} + {(- 1)}^{3}$

Step 1.18

Raise $- 1$ to the power of $3$ .

$6 + \frac{3}{4} - 1$

Step 2

Find the common denominator.

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Step 2.1

Write $6$ as a fraction with denominator $1$ .

$\frac{6}{1} + \frac{3}{4} - 1$

Step 2.2

Multiply $\frac{6}{1}$ by $\frac{4}{4}$ .

$\frac{6}{1} \cdot \frac{4}{4} + \frac{3}{4} - 1$

Step 2.3

Multiply $\frac{6}{1}$ by $\frac{4}{4}$ .

$\frac{6 \cdot 4}{4} + \frac{3}{4} - 1$

Step 2.4

Write $- 1$ as a fraction with denominator $1$ .

$\frac{6 \cdot 4}{4} + \frac{3}{4} + \frac{- 1}{1}$

Step 2.5

Multiply $\frac{- 1}{1}$ by $\frac{4}{4}$ .

$\frac{6 \cdot 4}{4} + \frac{3}{4} + \frac{- 1}{1} \cdot \frac{4}{4}$

Step 2.6

Multiply $\frac{- 1}{1}$ by $\frac{4}{4}$ .

$\frac{6 \cdot 4}{4} + \frac{3}{4} + \frac{- 1 \cdot 4}{4}$

Step 3

Combine the numerators over the common denominator.

$\frac{6 \cdot 4 + 3 - 1 \cdot 4}{4}$

Step 4

Simplify each term.

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Step 4.1

Multiply $6$ by $4$ .

$\frac{24 + 3 - 1 \cdot 4}{4}$

Step 4.2

Multiply $- 1$ by $4$ .

$\frac{24 + 3 - 4}{4}$

Step 5

Simplify by adding and subtracting.

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Step 5.1

Add $24$ and $3$ .

$\frac{27 - 4}{4}$

Step 5.2

Subtract $4$ from $27$ .

$\frac{23}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{23}{4}$

Decimal Form:

$5.75$

Mixed Number Form:

$5 \frac{3}{4}$