Trigonometry Examples

$\frac{7 \cos (7 π + i \sin (7 π))}{5 \cos (5 π + i \sin (5 π))}$

Step 1

Separate fractions.

$\frac{7}{5} \cdot \frac{\cos (7 π + i \sin (7 π))}{\cos (5 π + i \sin (5 π))}$

Step 2

Rewrite $\frac{\cos (7 π + i \sin (7 π))}{\cos (5 π + i \sin (5 π))}$ as a product.

$\frac{7}{5} (\cos (7 π + i \sin (7 π)) \frac{1}{\cos (5 π + i \sin (5 π))})$

Step 3

Write $\cos (7 π + i \sin (7 π))$ as a fraction with denominator $1$ .

$\frac{7}{5} (\frac{\cos (7 π + i \sin (7 π))}{1} \cdot \frac{1}{\cos (5 π + i \sin (5 π))})$

Step 4

Simplify.

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Step 4.1

Divide $\cos (7 π + i \sin (7 π))$ by $1$ .

$\frac{7}{5} (\cos (7 π + i \sin (7 π)) \frac{1}{\cos (5 π + i \sin (5 π))})$

Step 4.2

Convert from $\frac{1}{\cos (5 π + i \sin (5 π))}$ to $\sec (5 π + i \sin (5 π))$ .

$\frac{7}{5} (\cos (7 π + i \sin (7 π)) \sec (5 π + i \sin (5 π)))$

Step 5

Simplify each term.

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Step 5.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{7}{5} (\cos (7 π + i \sin (π)) \sec (5 π + i \sin (5 π)))$

Step 5.2

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

$\frac{7}{5} (\cos (7 π + i \sin (0)) \sec (5 π + i \sin (5 π)))$

Step 5.3

The exact value of $\sin (0)$ is $0$ .

$\frac{7}{5} (\cos (7 π + i \cdot 0) \sec (5 π + i \sin (5 π)))$

Step 5.4

Multiply $i$ by $0$ .

$\frac{7}{5} (\cos (7 π + 0) \sec (5 π + i \sin (5 π)))$

Step 6

Add $7 π$ and $0$ .

$\frac{7}{5} (\cos (7 π) \sec (5 π + i \sin (5 π)))$

Step 7

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{7}{5} (\cos (π) \sec (5 π + i \sin (5 π)))$

Step 8

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.

$\frac{7}{5} (- \cos (0) \sec (5 π + i \sin (5 π)))$

Step 9

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

$\frac{7}{5} (- 1 \cdot 1 \sec (5 π + i \sin (5 π)))$

Step 10

Multiply $- 1$ by $1$ .

$\frac{7}{5} (- \sec (5 π + i \sin (5 π)))$

Step 11

Simplify each term.

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Step 11.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{7}{5} (- \sec (5 π + i \sin (π)))$

Step 11.2

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

$\frac{7}{5} (- \sec (5 π + i \sin (0)))$

Step 11.3

The exact value of $\sin (0)$ is $0$ .

$\frac{7}{5} (- \sec (5 π + i \cdot 0))$

Step 11.4

Multiply $i$ by $0$ .

$\frac{7}{5} (- \sec (5 π + 0))$

Step 12

Add $5 π$ and $0$ .

$\frac{7}{5} (- \sec (5 π))$

Step 13

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{7}{5} (- \sec (π))$

Step 14

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.

$\frac{7}{5} (- - \sec (0))$

Step 15

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

$\frac{7}{5} (- (- 1 \cdot 1))$

Step 16

Multiply $- (- 1 \cdot 1)$ .

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Step 16.1

Multiply $- 1$ by $1$ .

$\frac{7}{5} (- - 1)$

Step 16.2

Multiply $- 1$ by $- 1$ .

$\frac{7}{5} \cdot 1$

Step 17

Multiply $\frac{7}{5}$ by $1$ .

$\frac{7}{5}$

Step 18

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{5}$

Decimal Form:

$1.4$

Mixed Number Form:

$1 \frac{2}{5}$