Trigonometry Examples

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

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

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

The exact value of

cos (60)

$\cos (60)$ is

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Step 1.2

Apply the product rule to

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Step 1.3

One to any power is one.

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

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

Step 1.4

Raise

2

$2$ to the power of

2

$2$ .

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

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

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)

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

Step 1.6

The exact value of

sec (30)

$\sec (30)$ is

\frac{2}{\sqrt{3}}

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

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

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

Step 1.7

Multiply

\frac{2}{\sqrt{3}}

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

\frac{\sqrt{3}}{\sqrt{3}}

$\frac{\sqrt{3}}{\sqrt{3}}$ .

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

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

Step 1.8

Combine and simplify the denominator.

Tap for more steps...

Step 1.8.1

Multiply

\frac{2}{\sqrt{3}}

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

\frac{\sqrt{3}}{\sqrt{3}}

$\frac{\sqrt{3}}{\sqrt{3}}$ .

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

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

Step 1.8.2

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

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

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

Step 1.8.3

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

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

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

Step 1.8.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 1.8.5

Add

1

$1$ and

1

$1$ .

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

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

Step 1.8.6

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

Tap for more steps...

Step 1.8.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{3}

$\sqrt{3}$ as

3^{\frac{1}{2}}

$3^{\frac{1}{2}}$ .

\frac{1}{4} + {⎛ ⎜ ⎜ ⎝ - \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} ⎞ ⎟ ⎟ ⎠}^{2} + {csc}^{2} (225)

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

Step 1.8.6.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 1.8.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 1.8.6.4

Cancel the common factor of

2

$2$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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

$3$ .

Tap for more steps...

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

$2$ .

Tap for more steps...

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

$9$ .

Tap for more steps...

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.

Tap for more steps...

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

$2$ .

Tap for more steps...

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

$2$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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:

$3 \frac{7}{12}$