Trigonometry Examples

$\sin (150) - \cos (330) \tan (150)$

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.

$\sin (30) - \cos (330) \tan (150)$

Step 1.2

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

$\frac{1}{2} - \cos (330) \tan (150)$

Step 1.3

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

$\frac{1}{2} - \cos (30) \tan (150)$

Step 1.4

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{1}{2} - \frac{\sqrt{3}}{2} \tan (150)$

Step 1.5

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 second quadrant.

$\frac{1}{2} - \frac{\sqrt{3}}{2} (- \tan (30))$

Step 1.6

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{1}{2} - \frac{\sqrt{3}}{2} (- \frac{\sqrt{3}}{3})$

Step 1.7

Multiply $- \frac{\sqrt{3}}{2} (- \frac{\sqrt{3}}{3})$ .

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Step 1.7.1

Multiply $- 1$ by $- 1$ .

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

Step 1.7.2

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

$\frac{1}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{3}$

Step 1.7.3

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{\sqrt{3}}{3}$ .

$\frac{1}{2} + \frac{\sqrt{3} \sqrt{3}}{2 \cdot 3}$

Step 1.7.4

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 1.7.5

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 1.7.6

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

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

Step 1.7.7

Add $1$ and $1$ .

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

Step 1.7.8

Multiply $2$ by $3$ .

$\frac{1}{2} + \frac{{\sqrt{3}}^{2}}{6}$

Step 1.8

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

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Step 1.8.1

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

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

Step 1.8.2

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

$\frac{1}{2} + \frac{3^{\frac{1}{2} \cdot 2}}{6}$

Step 1.8.3

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

$\frac{1}{2} + \frac{3^{\frac{2}{2}}}{6}$

Step 1.8.4

Cancel the common factor of $2$ .

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Step 1.8.4.1

Cancel the common factor.

$\frac{1}{2} + \frac{3^{\frac{2}{2}}}{6}$

Step 1.8.4.2

Rewrite the expression.

$\frac{1}{2} + \frac{3^{1}}{6}$

Step 1.8.5

Evaluate the exponent.

$\frac{1}{2} + \frac{3}{6}$

Step 1.9

Cancel the common factor of $3$ and $6$ .

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Step 1.9.1

Factor $3$ out of $3$ .

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

Step 1.9.2

Cancel the common factors.

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Step 1.9.2.1

Factor $3$ out of $6$ .

$\frac{1}{2} + \frac{3 \cdot 1}{3 \cdot 2}$

Step 1.9.2.2

Cancel the common factor.

$\frac{1}{2} + \frac{3 \cdot 1}{3 \cdot 2}$

Step 1.9.2.3

Rewrite the expression.

$\frac{1}{2} + \frac{1}{2}$

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{1 + 1}{2}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Add $1$ and $1$ .

$\frac{2}{2}$

Step 2.2.2

Divide $2$ by $2$ .

$1$