Trigonometry Examples

$\cos^{2} (60) + \sec^{2} (150) + \csc^{2} (225)$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

${(\frac{1}{2})}^{2} + \sec^{2} (150) + \csc^{2} (225)$

Step 1.2

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

$\frac{1^{2}}{2^{2}} + \sec^{2} (150) + \csc^{2} (225)$

Step 1.3

One to any power is one.

$\frac{1}{2^{2}} + \sec^{2} (150) + \csc^{2} (225)$

Step 1.4

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

$\frac{1}{4} + \sec^{2} (150) + \csc^{2} (225)$

Step 1.5

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

$\frac{1}{4} + {(- \sec (30))}^{2} + \csc^{2} (225)$

Step 1.6

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

$\frac{1}{4} + {(- \frac{2}{\sqrt{3}})}^{2} + \csc^{2} (225)$

Step 1.7

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

$\frac{1}{4} + {(- (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}))}^{2} + \csc^{2} (225)$

Step 1.8

Combine and simplify the denominator.

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Step 1.8.1

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

$\frac{1}{4} + {(- \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}})}^{2} + \csc^{2} (225)$

Step 1.8.2

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

$\frac{1}{4} + {(- \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}})}^{2} + \csc^{2} (225)$

Step 1.8.3

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

$\frac{1}{4} + {(- \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}})}^{2} + \csc^{2} (225)$

Step 1.8.4

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

$\frac{1}{4} + {(- \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}})}^{2} + \csc^{2} (225)$

Step 1.8.5

Add $1$ and $1$ .

$\frac{1}{4} + {(- \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}})}^{2} + \csc^{2} (225)$

Step 1.8.6

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

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Step 1.8.6.1

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

$\frac{1}{4} + {(- \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}})}^{2} + \csc^{2} (225)$

Step 1.8.6.2

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

$\frac{1}{4} + {(- \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}})}^{2} + \csc^{2} (225)$

Step 1.8.6.3

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

$\frac{1}{4} + {(- \frac{2 \sqrt{3}}{3^{\frac{2}{2}}})}^{2} + \csc^{2} (225)$

Step 1.8.6.4

Cancel the common factor of $2$ .

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Step 1.8.6.4.1

Cancel the common factor.

$\frac{1}{4} + {(- \frac{2 \sqrt{3}}{3^{\frac{2}{2}}})}^{2} + \csc^{2} (225)$

Step 1.8.6.4.2

Rewrite the expression.

$\frac{1}{4} + {(- \frac{2 \sqrt{3}}{3^{1}})}^{2} + \csc^{2} (225)$

Step 1.8.6.5

Evaluate the exponent.

$\frac{1}{4} + {(- \frac{2 \sqrt{3}}{3})}^{2} + \csc^{2} (225)$

Step 1.9

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.9.1

Apply the product rule to $- \frac{2 \sqrt{3}}{3}$ .

$\frac{1}{4} + {(- 1)}^{2} {(\frac{2 \sqrt{3}}{3})}^{2} + \csc^{2} (225)$

Step 1.9.2

Apply the product rule to $\frac{2 \sqrt{3}}{3}$ .

$\frac{1}{4} + {(- 1)}^{2} \frac{{(2 \sqrt{3})}^{2}}{3^{2}} + \csc^{2} (225)$

Step 1.9.3

Apply the product rule to $2 \sqrt{3}$ .

$\frac{1}{4} + {(- 1)}^{2} \frac{2^{2} {\sqrt{3}}^{2}}{3^{2}} + \csc^{2} (225)$

Step 1.10

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

$\frac{1}{4} + 1 \frac{2^{2} {\sqrt{3}}^{2}}{3^{2}} + \csc^{2} (225)$

Step 1.11

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

$\frac{1}{4} + \frac{2^{2} {\sqrt{3}}^{2}}{3^{2}} + \csc^{2} (225)$

Step 1.12

Simplify the numerator.

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Step 1.12.1

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

$\frac{1}{4} + \frac{4 {\sqrt{3}}^{2}}{3^{2}} + \csc^{2} (225)$

Step 1.12.2

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

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Step 1.12.2.1

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

$\frac{1}{4} + \frac{4 {(3^{\frac{1}{2}})}^{2}}{3^{2}} + \csc^{2} (225)$

Step 1.12.2.2

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

$\frac{1}{4} + \frac{4 \cdot 3^{\frac{1}{2} \cdot 2}}{3^{2}} + \csc^{2} (225)$

Step 1.12.2.3

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

$\frac{1}{4} + \frac{4 \cdot 3^{\frac{2}{2}}}{3^{2}} + \csc^{2} (225)$

Step 1.12.2.4

Cancel the common factor of $2$ .

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Step 1.12.2.4.1

Cancel the common factor.

$\frac{1}{4} + \frac{4 \cdot 3^{\frac{2}{2}}}{3^{2}} + \csc^{2} (225)$

Step 1.12.2.4.2

Rewrite the expression.

$\frac{1}{4} + \frac{4 \cdot 3^{1}}{3^{2}} + \csc^{2} (225)$

Step 1.12.2.5

Evaluate the exponent.

$\frac{1}{4} + \frac{4 \cdot 3}{3^{2}} + \csc^{2} (225)$

Step 1.13

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

$\frac{1}{4} + \frac{4 \cdot 3}{9} + \csc^{2} (225)$

Step 1.14

Multiply $4$ by $3$ .

$\frac{1}{4} + \frac{12}{9} + \csc^{2} (225)$

Step 1.15

Cancel the common factor of $12$ and $9$ .

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Step 1.15.1

Factor $3$ out of $12$ .

$\frac{1}{4} + \frac{3 (4)}{9} + \csc^{2} (225)$

Step 1.15.2

Cancel the common factors.

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Step 1.15.2.1

Factor $3$ out of $9$ .

$\frac{1}{4} + \frac{3 \cdot 4}{3 \cdot 3} + \csc^{2} (225)$

Step 1.15.2.2

Cancel the common factor.

$\frac{1}{4} + \frac{3 \cdot 4}{3 \cdot 3} + \csc^{2} (225)$

Step 1.15.2.3

Rewrite the expression.

$\frac{1}{4} + \frac{4}{3} + \csc^{2} (225)$

Step 1.16

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

$\frac{1}{4} + \frac{4}{3} + {(- \csc (45))}^{2}$

Step 1.17

The exact value of $\csc (45)$ is $\sqrt{2}$ .

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

Step 1.18

Apply the product rule to $- \sqrt{2}$ .

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

Step 1.19

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

$\frac{1}{4} + \frac{4}{3} + 1 {\sqrt{2}}^{2}$

Step 1.20

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

$\frac{1}{4} + \frac{4}{3} + {\sqrt{2}}^{2}$

Step 1.21

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

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Step 1.21.1

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

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

Step 1.21.2

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

$\frac{1}{4} + \frac{4}{3} + 2^{\frac{1}{2} \cdot 2}$

Step 1.21.3

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

$\frac{1}{4} + \frac{4}{3} + 2^{\frac{2}{2}}$

Step 1.21.4

Cancel the common factor of $2$ .

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Step 1.21.4.1

Cancel the common factor.

$\frac{1}{4} + \frac{4}{3} + 2^{\frac{2}{2}}$

Step 1.21.4.2

Rewrite the expression.

$\frac{1}{4} + \frac{4}{3} + 2^{1}$

Step 1.21.5

Evaluate the exponent.

$\frac{1}{4} + \frac{4}{3} + 2$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{1}{4}$ by $\frac{3}{3}$ .

$\frac{1}{4} \cdot \frac{3}{3} + \frac{4}{3} + 2$

Step 2.2

Multiply $\frac{1}{4}$ by $\frac{3}{3}$ .

$\frac{3}{4 \cdot 3} + \frac{4}{3} + 2$

Step 2.3

Multiply $\frac{4}{3}$ by $\frac{4}{4}$ .

$\frac{3}{4 \cdot 3} + \frac{4}{3} \cdot \frac{4}{4} + 2$

Step 2.4

Multiply $\frac{4}{3}$ by $\frac{4}{4}$ .

$\frac{3}{4 \cdot 3} + \frac{4 \cdot 4}{3 \cdot 4} + 2$

Step 2.5

Write $2$ as a fraction with denominator $1$ .

$\frac{3}{4 \cdot 3} + \frac{4 \cdot 4}{3 \cdot 4} + \frac{2}{1}$

Step 2.6

Multiply $\frac{2}{1}$ by $\frac{12}{12}$ .

$\frac{3}{4 \cdot 3} + \frac{4 \cdot 4}{3 \cdot 4} + \frac{2}{1} \cdot \frac{12}{12}$

Step 2.7

Multiply $\frac{2}{1}$ by $\frac{12}{12}$ .

$\frac{3}{4 \cdot 3} + \frac{4 \cdot 4}{3 \cdot 4} + \frac{2 \cdot 12}{12}$

Step 2.8

Reorder the factors of $4 \cdot 3$ .

$\frac{3}{3 \cdot 4} + \frac{4 \cdot 4}{3 \cdot 4} + \frac{2 \cdot 12}{12}$

Step 2.9

Multiply $3$ by $4$ .

$\frac{3}{12} + \frac{4 \cdot 4}{3 \cdot 4} + \frac{2 \cdot 12}{12}$

Step 2.10

Multiply $3$ by $4$ .

$\frac{3}{12} + \frac{4 \cdot 4}{12} + \frac{2 \cdot 12}{12}$

Step 3

Combine the numerators over the common denominator.

$\frac{3 + 4 \cdot 4 + 2 \cdot 12}{12}$

Step 4

Simplify each term.

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Step 4.1

Multiply $4$ by $4$ .

$\frac{3 + 16 + 2 \cdot 12}{12}$

Step 4.2

Multiply $2$ by $12$ .

$\frac{3 + 16 + 24}{12}$

Step 5

Simplify by adding numbers.

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Step 5.1

Add $3$ and $16$ .

$\frac{19 + 24}{12}$

Step 5.2

Add $19$ and $24$ .

$\frac{43}{12}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{43}{12}$

Decimal Form:

$3.58\overline{3}$

Mixed Number Form:

$3 \frac{7}{12}$