Trigonometry Examples

\sin^{2} (30) \cdot \cos (180) + \tan (0) - \cot (- 45)

$\sin^{2} (30) \cdot \cos (180) + \tan (0) - \cot (- 45)$

Step 1

Simplify each term.

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Step 1.1

The exact value of

\sin (30)

$\sin (30)$ is

\frac{1}{2}

$\frac{1}{2}$ .

{(\frac{1}{2})}^{2} \cdot \cos (180) + \tan (0) - \cot (- 45)

${(\frac{1}{2})}^{2} \cdot \cos (180) + \tan (0) - \cot (- 45)$

Step 1.2

Apply the product rule to

\frac{1}{2}

$\frac{1}{2}$ .

\frac{1^{2}}{2^{2}} \cdot \cos (180) + \tan (0) - \cot (- 45)

$\frac{1^{2}}{2^{2}} \cdot \cos (180) + \tan (0) - \cot (- 45)$

Step 1.3

One to any power is one.

\frac{1}{2^{2}} \cdot \cos (180) + \tan (0) - \cot (- 45)

$\frac{1}{2^{2}} \cdot \cos (180) + \tan (0) - \cot (- 45)$

Step 1.4

Raise

2

$2$ to the power of

2

$2$ .

\frac{1}{4} \cdot \cos (180) + \tan (0) - \cot (- 45)

$\frac{1}{4} \cdot \cos (180) + \tan (0) - \cot (- 45)$

Step 1.5

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.

\frac{1}{4} \cdot (- \cos (0)) + \tan (0) - \cot (- 45)

$\frac{1}{4} \cdot (- \cos (0)) + \tan (0) - \cot (- 45)$

Step 1.6

The exact value of

\cos (0)

$\cos (0)$ is

1

$1$ .

\frac{1}{4} \cdot (- 1 \cdot 1) + \tan (0) - \cot (- 45)

$\frac{1}{4} \cdot (- 1 \cdot 1) + \tan (0) - \cot (- 45)$

Step 1.7

Multiply

- 1

$- 1$ by

1

$1$ .

\frac{1}{4} \cdot - 1 + \tan (0) - \cot (- 45)

$\frac{1}{4} \cdot - 1 + \tan (0) - \cot (- 45)$

Step 1.8

Combine

\frac{1}{4}

$\frac{1}{4}$ and

- 1

$- 1$ .

\frac{- 1}{4} + \tan (0) - \cot (- 45)

$\frac{- 1}{4} + \tan (0) - \cot (- 45)$

Step 1.9

Move the negative in front of the fraction.

- \frac{1}{4} + \tan (0) - \cot (- 45)

$- \frac{1}{4} + \tan (0) - \cot (- 45)$

Step 1.10

The exact value of

\tan (0)

$\tan (0)$ is

0

$0$ .

- \frac{1}{4} + 0 - \cot (- 45)

$- \frac{1}{4} + 0 - \cot (- 45)$

Step 1.11

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

- \frac{1}{4} + 0 - - \cot (45)

$- \frac{1}{4} + 0 - - \cot (45)$

Step 1.12

The exact value of

\cot (45)

$\cot (45)$ is

1

$1$ .

- \frac{1}{4} + 0 - (- 1 \cdot 1)

$- \frac{1}{4} + 0 - (- 1 \cdot 1)$

Step 1.13

Multiply

- (- 1 \cdot 1)

$- (- 1 \cdot 1)$ .

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Step 1.13.1

Multiply

- 1

$- 1$ by

1

$1$ .

- \frac{1}{4} + 0 - - 1

$- \frac{1}{4} + 0 - - 1$

Step 1.13.2

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

- \frac{1}{4} + 0 + 1

$- \frac{1}{4} + 0 + 1$

- \frac{1}{4} + 0 + 1

$- \frac{1}{4} + 0 + 1$

- \frac{1}{4} + 0 + 1

$- \frac{1}{4} + 0 + 1$

Step 2

Simplify the expression.

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Step 2.1

Add

- \frac{1}{4}

$- \frac{1}{4}$ and

0

$0$ .

- \frac{1}{4} + 1

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

Step 2.2

Write

1

$1$ as a fraction with a common denominator.

- \frac{1}{4} + \frac{4}{4}

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

Step 2.3

Combine the numerators over the common denominator.

\frac{- 1 + 4}{4}

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

Step 2.4

Add

- 1

$- 1$ and

4

$4$ .

\frac{3}{4}

$\frac{3}{4}$

\frac{3}{4}

$\frac{3}{4}$

Step 3

The result can be shown in multiple forms.

Exact Form:

\frac{3}{4}

$\frac{3}{4}$

Decimal Form:

0.75

$0.75$