Trigonometry Examples

$\frac{1 - \cos (x)}{\sin (x)} = \frac{\sin (1)}{1 + \cos (x)}$

Step 1

Divide each term in the equation by $\cos (x)$ .

$\frac{\frac{1 - \cos (x)}{\sin (x)}}{\cos (x)} = \frac{\frac{\sin (1)}{1 + \cos (x)}}{\cos (x)}$

Step 2

Separate fractions.

$\frac{\frac{1 - \cos (x)}{\sin (x)}}{1} \cdot \frac{1}{\cos (x)} = \frac{\frac{\sin (1)}{1 + \cos (x)}}{\cos (x)}$

Step 3

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{\frac{1 - \cos (x)}{\sin (x)}}{1} \cdot \sec (x) = \frac{\frac{\sin (1)}{1 + \cos (x)}}{\cos (x)}$

Step 4

Divide $\frac{1 - \cos (x)}{\sin (x)}$ by $1$ .

$\frac{1 - \cos (x)}{\sin (x)} \cdot \sec (x) = \frac{\frac{\sin (1)}{1 + \cos (x)}}{\cos (x)}$

Step 5

Combine $\frac{1 - \cos (x)}{\sin (x)}$ and $\sec (x)$ .

$\frac{(1 - \cos (x)) \sec (x)}{\sin (x)} = \frac{\frac{\sin (1)}{1 + \cos (x)}}{\cos (x)}$

Step 6

Separate fractions.

$\frac{1 - \cos (x)}{1} \cdot \frac{\sec (x)}{\sin (x)} = \frac{\frac{\sin (1)}{1 + \cos (x)}}{\cos (x)}$

Step 7

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{1 - \cos (x)}{1} \cdot \frac{\frac{1}{\cos (x)}}{\sin (x)} = \frac{\frac{\sin (1)}{1 + \cos (x)}}{\cos (x)}$

Step 8

Rewrite $\frac{\frac{1}{\cos (x)}}{\sin (x)}$ as a product.

$\frac{1 - \cos (x)}{1} \cdot (\frac{1}{\cos (x)} \cdot \frac{1}{\sin (x)}) = \frac{\frac{\sin (1)}{1 + \cos (x)}}{\cos (x)}$

Step 9

Simplify.

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Step 9.1

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{1 - \cos (x)}{1} \cdot (\sec (x) (\frac{1}{\sin (x)})) = \frac{\frac{\sin (1)}{1 + \cos (x)}}{\cos (x)}$

Step 9.2

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\frac{1 - \cos (x)}{1} \cdot (\sec (x) \csc (x)) = \frac{\frac{\sin (1)}{1 + \cos (x)}}{\cos (x)}$

Step 10

Simplify by multiplying through.

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Step 10.1

Divide $1 - \cos (x)$ by $1$ .

$(1 - \cos (x)) (\sec (x) \csc (x)) = \frac{\frac{\sin (1)}{1 + \cos (x)}}{\cos (x)}$

Step 10.2

Apply the distributive property.

$1 (\sec (x) \csc (x)) - \cos (x) (\sec (x) \csc (x)) = \frac{\frac{\sin (1)}{1 + \cos (x)}}{\cos (x)}$

Step 10.3

Multiply $\sec (x) \csc (x)$ by $1$ .

$\sec (x) \csc (x) - \cos (x) (\sec (x) \csc (x)) = \frac{\frac{\sin (1)}{1 + \cos (x)}}{\cos (x)}$

Step 11

Simplify each term.

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Step 11.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 11.1.1

Add parentheses.

$\sec (x) \csc (x) - \cos (x) \sec (x) \csc (x) = \frac{\frac{\sin (1)}{1 + \cos (x)}}{\cos (x)}$

Step 11.1.2

Reorder $\cos (x)$ and $\sec (x)$ .

$\sec (x) \csc (x) - \sec (x) \cos (x) \csc (x) = \frac{\frac{\sin (1)}{1 + \cos (x)}}{\cos (x)}$

Step 11.1.3

Rewrite $- \cos (x) \sec (x)$ in terms of sines and cosines.

$\sec (x) \csc (x) - \frac{1}{\cos (x)} \cdot \cos (x) \csc (x) = \frac{\frac{\sin (1)}{1 + \cos (x)}}{\cos (x)}$

Step 11.1.4

Cancel the common factors.

$\sec (x) \csc (x) - 1 \cdot (1 \csc (x)) = \frac{\frac{\sin (1)}{1 + \cos (x)}}{\cos (x)}$

Step 11.2

Multiply $- 1$ by $1$ .

$\sec (x) \csc (x) - \csc (x) = \frac{\frac{\sin (1)}{1 + \cos (x)}}{\cos (x)}$

Step 12

Separate fractions.

$\sec (x) \csc (x) - \csc (x) = \frac{\frac{\sin (1)}{1 + \cos (x)}}{1} \cdot \frac{1}{\cos (x)}$

Step 13

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x) \csc (x) - \csc (x) = \frac{\frac{\sin (1)}{1 + \cos (x)}}{1} \cdot \sec (x)$

Step 14

Divide $\frac{\sin (1)}{1 + \cos (x)}$ by $1$ .

$\sec (x) \csc (x) - \csc (x) = \frac{\sin (1)}{1 + \cos (x)} \cdot \sec (x)$

Step 15

Evaluate $\sin (1)$ .

$\sec (x) \csc (x) - \csc (x) = \frac{0.0174524}{1 + \cos (x)} \cdot \sec (x)$

Step 16

Combine $\frac{0.0174524}{1 + \cos (x)}$ and $\sec (x)$ .

$\sec (x) \csc (x) - \csc (x) = \frac{0.0174524 \sec (x)}{1 + \cos (x)}$

Step 17

Multiply both sides by $1 + \cos (x)$ .

$(\sec (x) \csc (x) - \csc (x)) (1 + \cos (x)) = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18

Simplify.

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Step 18.1

Simplify the left side.

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Step 18.1.1

Simplify $(\sec (x) \csc (x) - \csc (x)) (1 + \cos (x))$ .

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Step 18.1.1.1

Simplify each term.

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Step 18.1.1.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$(\frac{1}{\cos (x)} \csc (x) - \csc (x)) (1 + \cos (x)) = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.1.1.1.2

Rewrite $\csc (x)$ in terms of sines and cosines.

$(\frac{1}{\cos (x)} \cdot \frac{1}{\sin (x)} - \csc (x)) (1 + \cos (x)) = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.1.1.1.3

Multiply $\frac{1}{\cos (x)}$ by $\frac{1}{\sin (x)}$ .

$(\frac{1}{\cos (x) \sin (x)} - \csc (x)) (1 + \cos (x)) = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.1.1.1.4

Rewrite $\csc (x)$ in terms of sines and cosines.

$(\frac{1}{\cos (x) \sin (x)} - \frac{1}{\sin (x)}) (1 + \cos (x)) = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.1.1.2

Expand $(\frac{1}{\cos (x) \sin (x)} - \frac{1}{\sin (x)}) (1 + \cos (x))$ using the FOIL Method.

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Step 18.1.1.2.1

Apply the distributive property.

$\frac{1}{\cos (x) \sin (x)} (1 + \cos (x)) - \frac{1}{\sin (x)} (1 + \cos (x)) = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.1.1.2.2

Apply the distributive property.

$\frac{1}{\cos (x) \sin (x)} \cdot 1 + \frac{1}{\cos (x) \sin (x)} \cos (x) - \frac{1}{\sin (x)} (1 + \cos (x)) = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.1.1.2.3

Apply the distributive property.

$\frac{1}{\cos (x) \sin (x)} \cdot 1 + \frac{1}{\cos (x) \sin (x)} \cos (x) - \frac{1}{\sin (x)} \cdot 1 - \frac{1}{\sin (x)} \cos (x) = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.1.1.3

Simplify and combine like terms.

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Step 18.1.1.3.1

Simplify each term.

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Step 18.1.1.3.1.1

Multiply $\frac{1}{\cos (x) \sin (x)}$ by $1$ .

$\frac{1}{\cos (x) \sin (x)} + \frac{1}{\cos (x) \sin (x)} \cos (x) - \frac{1}{\sin (x)} \cdot 1 - \frac{1}{\sin (x)} \cos (x) = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.1.1.3.1.2

Cancel the common factor of $\cos (x)$ .

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Step 18.1.1.3.1.2.1

Factor $\cos (x)$ out of $\cos (x) \sin (x)$ .

$\frac{1}{\cos (x) \sin (x)} + \frac{1}{\cos (x) (\sin (x))} \cos (x) - \frac{1}{\sin (x)} \cdot 1 - \frac{1}{\sin (x)} \cos (x) = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.1.1.3.1.2.2

Cancel the common factor.

$\frac{1}{\cos (x) \sin (x)} + \frac{1}{\cos (x) \sin (x)} \cos (x) - \frac{1}{\sin (x)} \cdot 1 - \frac{1}{\sin (x)} \cos (x) = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.1.1.3.1.2.3

Rewrite the expression.

$\frac{1}{\cos (x) \sin (x)} + \frac{1}{\sin (x)} - \frac{1}{\sin (x)} \cdot 1 - \frac{1}{\sin (x)} \cos (x) = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.1.1.3.1.3

Multiply $- 1$ by $1$ .

$\frac{1}{\cos (x) \sin (x)} + \frac{1}{\sin (x)} - \frac{1}{\sin (x)} - \frac{1}{\sin (x)} \cos (x) = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.1.1.3.1.4

Combine $\cos (x)$ and $\frac{1}{\sin (x)}$ .

$\frac{1}{\cos (x) \sin (x)} + \frac{1}{\sin (x)} - \frac{1}{\sin (x)} - \frac{\cos (x)}{\sin (x)} = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.1.1.3.2

Subtract $\frac{1}{\sin (x)}$ from $\frac{1}{\sin (x)}$ .

$\frac{1}{\cos (x) \sin (x)} + 0 - \frac{\cos (x)}{\sin (x)} = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.1.1.3.3

Add $\frac{1}{\cos (x) \sin (x)}$ and $0$ .

$\frac{1}{\cos (x) \sin (x)} - \frac{\cos (x)}{\sin (x)} = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.1.1.4

Simplify each term.

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Step 18.1.1.4.1

Separate fractions.

$\frac{1}{\sin (x)} \cdot \frac{1}{\cos (x)} - \frac{\cos (x)}{\sin (x)} = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.1.1.4.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{1}{\sin (x)} \sec (x) - \frac{\cos (x)}{\sin (x)} = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.1.1.4.3

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\csc (x) \sec (x) - \frac{\cos (x)}{\sin (x)} = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.1.1.4.4

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\csc (x) \sec (x) - \cot (x) = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.2

Simplify the right side.

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Step 18.2.1

Simplify $\frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$ .

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Step 18.2.1.1

Cancel the common factor of $1 + \cos (x)$ .

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Step 18.2.1.1.1

Cancel the common factor.

$\csc (x) \sec (x) - \cot (x) = \frac{0.0174524 \sec (x)}{1 + \cos (x)} (1 + \cos (x))$

Step 18.2.1.1.2

Rewrite the expression.

$\csc (x) \sec (x) - \cot (x) = 0.0174524 \sec (x)$

Step 18.2.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$\csc (x) \sec (x) - \cot (x) = 0.0174524 \frac{1}{\cos (x)}$

Step 18.2.1.3

Combine $0.0174524$ and $\frac{1}{\cos (x)}$ .

$\csc (x) \sec (x) - \cot (x) = \frac{0.0174524}{\cos (x)}$

Step 18.2.1.4

Separate fractions.

$\csc (x) \sec (x) - \cot (x) = \frac{0.0174524}{1} \cdot \frac{1}{\cos (x)}$

Step 18.2.1.5

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\csc (x) \sec (x) - \cot (x) = \frac{0.0174524}{1} \sec (x)$

Step 18.2.1.6

Divide $0.0174524$ by $1$ .

$\csc (x) \sec (x) - \cot (x) = 0.0174524 \sec (x)$

Step 19

Solve for $x$ .

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Step 19.1

Simplify the left side.

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Step 19.1.1

Simplify each term.

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Step 19.1.1.1

Rewrite $\csc (x)$ in terms of sines and cosines.

$\frac{1}{\sin (x)} \sec (x) - \cot (x) = 0.0174524 \sec (x)$

Step 19.1.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{1}{\sin (x)} \cdot \frac{1}{\cos (x)} - \cot (x) = 0.0174524 \sec (x)$

Step 19.1.1.3

Multiply $\frac{1}{\sin (x)}$ by $\frac{1}{\cos (x)}$ .

$\frac{1}{\sin (x) \cos (x)} - \cot (x) = 0.0174524 \sec (x)$

Step 19.1.1.4

Rewrite $\cot (x)$ in terms of sines and cosines.

$\frac{1}{\sin (x) \cos (x)} - \frac{\cos (x)}{\sin (x)} = 0.0174524 \sec (x)$

Step 19.2

Simplify the right side.

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Step 19.2.1

Simplify $0.0174524 \sec (x)$ .

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Step 19.2.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{1}{\sin (x) \cos (x)} - \frac{\cos (x)}{\sin (x)} = 0.0174524 \frac{1}{\cos (x)}$

Step 19.2.1.2

Combine $0.0174524$ and $\frac{1}{\cos (x)}$ .

$\frac{1}{\sin (x) \cos (x)} - \frac{\cos (x)}{\sin (x)} = \frac{0.0174524}{\cos (x)}$

Step 19.3

Multiply both sides of the equation by $\sin (x)$ .

$\sin (x) (\frac{1}{\sin (x) \cos (x)} - \frac{\cos (x)}{\sin (x)}) = \sin (x) \frac{0.0174524}{\cos (x)}$

Step 19.4

Apply the distributive property.

$\sin (x) \frac{1}{\sin (x) \cos (x)} + \sin (x) (- \frac{\cos (x)}{\sin (x)}) = \sin (x) \frac{0.0174524}{\cos (x)}$

Step 19.5

Cancel the common factor of $\sin (x)$ .

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Step 19.5.1

Factor $\sin (x)$ out of $\sin (x) \cos (x)$ .

$\sin (x) \frac{1}{\sin (x) (\cos (x))} + \sin (x) (- \frac{\cos (x)}{\sin (x)}) = \sin (x) \frac{0.0174524}{\cos (x)}$

Step 19.5.2

Cancel the common factor.

$\sin (x) \frac{1}{\sin (x) \cos (x)} + \sin (x) (- \frac{\cos (x)}{\sin (x)}) = \sin (x) \frac{0.0174524}{\cos (x)}$

Step 19.5.3

Rewrite the expression.

$\frac{1}{\cos (x)} + \sin (x) (- \frac{\cos (x)}{\sin (x)}) = \sin (x) \frac{0.0174524}{\cos (x)}$

Step 19.6

Rewrite using the commutative property of multiplication.

$\frac{1}{\cos (x)} - \sin (x) \frac{\cos (x)}{\sin (x)} = \sin (x) \frac{0.0174524}{\cos (x)}$

Step 19.7

Cancel the common factor of $\sin (x)$ .

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Step 19.7.1

Factor $\sin (x)$ out of $- \sin (x)$ .

$\frac{1}{\cos (x)} + \sin (x) \cdot - 1 \frac{\cos (x)}{\sin (x)} = \sin (x) \frac{0.0174524}{\cos (x)}$

Step 19.7.2

Cancel the common factor.

$\frac{1}{\cos (x)} + \sin (x) \cdot - 1 \frac{\cos (x)}{\sin (x)} = \sin (x) \frac{0.0174524}{\cos (x)}$

Step 19.7.3

Rewrite the expression.

$\frac{1}{\cos (x)} - \cos (x) = \sin (x) \frac{0.0174524}{\cos (x)}$

Step 19.8

Combine $\sin (x)$ and $\frac{0.0174524}{\cos (x)}$ .

$\frac{1}{\cos (x)} - \cos (x) = \frac{\sin (x) \cdot 0.0174524}{\cos (x)}$

Step 19.9

Move $0.0174524$ to the left of $\sin (x)$ .

$\frac{1}{\cos (x)} - \cos (x) = \frac{0.0174524 \sin (x)}{\cos (x)}$

Step 19.10

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) (\frac{1}{\cos (x)} - \cos (x)) = \cos (x) \frac{0.0174524 \sin (x)}{\cos (x)}$

Step 19.11

Apply the distributive property.

$\cos (x) \frac{1}{\cos (x)} + \cos (x) (- \cos (x)) = \cos (x) \frac{0.0174524 \sin (x)}{\cos (x)}$

Step 19.12

Cancel the common factor of $\cos (x)$ .

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Step 19.12.1

Cancel the common factor.

$\cos (x) \frac{1}{\cos (x)} + \cos (x) (- \cos (x)) = \cos (x) \frac{0.0174524 \sin (x)}{\cos (x)}$

Step 19.12.2

Rewrite the expression.

$1 + \cos (x) (- \cos (x)) = \cos (x) \frac{0.0174524 \sin (x)}{\cos (x)}$

Step 19.13

Rewrite using the commutative property of multiplication.

$1 - \cos (x) \cos (x) = \cos (x) \frac{0.0174524 \sin (x)}{\cos (x)}$

Step 19.14

Multiply $- \cos (x) \cos (x)$ .

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Step 19.14.1

Raise $\cos (x)$ to the power of $1$ .

$1 - (\cos^{1} (x) \cos (x)) = \cos (x) \frac{0.0174524 \sin (x)}{\cos (x)}$

Step 19.14.2

Raise $\cos (x)$ to the power of $1$ .

$1 - (\cos^{1} (x) \cos^{1} (x)) = \cos (x) \frac{0.0174524 \sin (x)}{\cos (x)}$

Step 19.14.3

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

$1 - {\cos (x)}^{1 + 1} = \cos (x) \frac{0.0174524 \sin (x)}{\cos (x)}$

Step 19.14.4

Add $1$ and $1$ .

$1 - \cos^{2} (x) = \cos (x) \frac{0.0174524 \sin (x)}{\cos (x)}$

Step 19.15

Apply pythagorean identity.

$\sin^{2} (x) = \cos (x) \frac{0.0174524 \sin (x)}{\cos (x)}$

Step 19.16

Cancel the common factor of $\cos (x)$ .

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Step 19.16.1

Cancel the common factor.

$\sin^{2} (x) = \cos (x) \frac{0.0174524 \sin (x)}{\cos (x)}$

Step 19.16.2

Rewrite the expression.

$\sin^{2} (x) = 0.0174524 \sin (x)$

Step 19.17

Subtract $0.0174524 \sin (x)$ from both sides of the equation.

$\sin^{2} (x) - 0.0174524 \sin (x) = 0$

Step 19.18

Factor $\sin (x)$ out of $\sin^{2} (x) - 0.0174524 \sin (x)$ .

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Step 19.18.1

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

$\sin (x) \sin (x) - 0.0174524 \sin (x) = 0$

Step 19.18.2

Factor $\sin (x)$ out of $- 0.0174524 \sin (x)$ .

$\sin (x) \sin (x) + \sin (x) \cdot - 0.0174524 = 0$

Step 19.18.3

Factor $\sin (x)$ out of $\sin (x) \sin (x) + \sin (x) \cdot - 0.0174524$ .

$\sin (x) (\sin (x) - 0.0174524) = 0$

Step 19.19

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\sin (x) = 0$

$\sin (x) - 0.0174524 = 0$

Step 19.20

Set $\sin (x)$ equal to $0$ and solve for $x$ .

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Step 19.20.1

Set $\sin (x)$ equal to $0$ .

$\sin (x) = 0$

Step 19.20.2

Solve $\sin (x) = 0$ for $x$ .

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Step 19.20.2.1

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$x = \arcsin (0)$

Step 19.20.2.2

Simplify the right side.

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Step 19.20.2.2.1

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

$x = 0$

Step 19.20.2.3

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$x = 180 - 0$

Step 19.20.2.4

Subtract $0$ from $180$ .

$x = 180$

Step 19.20.2.5

Find the period of $\sin (x)$ .

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Step 19.20.2.5.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 19.20.2.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{360}{| 1 |}$

Step 19.20.2.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{360}{1}$

Step 19.20.2.5.4

Divide $360$ by $1$ .

$360$

Step 19.20.2.6

The period of the $\sin (x)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$x = 360 n, 180 + 360 n$ , for any integer $n$

Step 19.21

Set $\sin (x) - 0.0174524$ equal to $0$ and solve for $x$ .

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Step 19.21.1

Set $\sin (x) - 0.0174524$ equal to $0$ .

$\sin (x) - 0.0174524 = 0$

Step 19.21.2

Solve $\sin (x) - 0.0174524 = 0$ for $x$ .

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Step 19.21.2.1

Add $0.0174524$ to both sides of the equation.

$\sin (x) = 0.0174524$

Step 19.21.2.2

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$x = \arcsin (0.0174524)$

Step 19.21.2.3

Simplify the right side.

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Step 19.21.2.3.1

Evaluate $\arcsin (0.0174524)$ .

$x = 1$

Step 19.21.2.4

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$x = 180 - 1$

Step 19.21.2.5

Subtract $1$ from $180$ .

$x = 179$

Step 19.21.2.6

Find the period of $\sin (x)$ .

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Step 19.21.2.6.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 19.21.2.6.2

Replace $b$ with $1$ in the formula for period.

$\frac{360}{| 1 |}$

Step 19.21.2.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{360}{1}$

Step 19.21.2.6.4

Divide $360$ by $1$ .

$360$

Step 19.21.2.7

The period of the $\sin (x)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$x = 1 + 360 n, 179 + 360 n$ , for any integer $n$

Step 19.22

The final solution is all the values that make $\sin (x) (\sin (x) - 0.0174524) = 0$ true.

$x = 360 n, 180 + 360 n, 1 + 360 n, 179 + 360 n$ , for any integer $n$

Step 20

Consolidate $360 n$ and $180 + 360 n$ to $180 n$ .

$x = 180 n, 1 + 360 n, 179 + 360 n$ , for any integer $n$

Step 21

Exclude the solutions that do not make $\frac{1 - \cos (x)}{\sin (x)} = \frac{\sin (1)}{1 + \cos (x)}$ true.

$x = 1 + 360 n, 179 + 360 n$ , for any integer $n$