Trigonometry Examples

$\sin (y) + \cos (x) = 1$

Step 1

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

$\sin (y) = 1 - \cos (x)$

Step 2

Rewrite the equation as $1 - \cos (x) = \sin (y)$ .

$1 - \cos (x) = \sin (y)$

Step 3

Subtract $1$ from both sides of the equation.

$- \cos (x) = \sin (y) - 1$

Step 4

Divide each term in $- \cos (x) = \sin (y) - 1$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- \cos (x) = \sin (y) - 1$ by $- 1$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

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

Step 4.2.2

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

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Move the negative one from the denominator of $\frac{\sin (y)}{- 1}$ .

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

Step 4.3.1.2

Rewrite $- 1 \cdot \sin (y)$ as $- \sin (y)$ .

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

Step 4.3.1.3

Divide $- 1$ by $- 1$ .

$\cos (x) = - \sin (y) + 1$

Step 5

Take the inverse cosine of both sides of the equation to extract $y$ from inside the cosine.

$x = \arccos (- \sin (y) + 1)$

Step 6

Rewrite the equation as $\arccos (- \sin (y) + 1) = x$ .

$\arccos (- \sin (y) + 1) = x$

Step 7

Take the inverse arccosine of both sides of the equation to extract $\sin (y)$ from inside the arccosine.

$- \sin (y) + 1 = \cos (x)$

Step 8

Subtract $1$ from both sides of the equation.

$- \sin (y) = \cos (x) - 1$

Step 9

Divide each term in $- \sin (y) = \cos (x) - 1$ by $- 1$ and simplify.

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Step 9.1

Divide each term in $- \sin (y) = \cos (x) - 1$ by $- 1$ .

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

Step 9.2

Simplify the left side.

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Step 9.2.1

Dividing two negative values results in a positive value.

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

Step 9.2.2

Divide $\sin (y)$ by $1$ .

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

Step 9.3

Simplify the right side.

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Step 9.3.1

Simplify each term.

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Step 9.3.1.1

Move the negative one from the denominator of $\frac{\cos (x)}{- 1}$ .

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

Step 9.3.1.2

Rewrite $- 1 \cdot \cos (x)$ as $- \cos (x)$ .

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

Step 9.3.1.3

Divide $- 1$ by $- 1$ .

$\sin (y) = - \cos (x) + 1$

Step 10

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

$y = \arcsin (- \cos (x) + 1)$