Trigonometry Examples

$\cos (π - x) + \sin (\frac{π}{2} + x) = 0$

Step 1

Start on the left side.

$\cos (π - x) + \sin (\frac{π}{2} + x)$

Step 2

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

$\cos (π) \cos (- x) - \sin (π) \sin (- x) + \sin (\frac{π}{2} + x)$

Step 3

Apply the sum of angles identity.

$\cos (π) \cos (- x) - \sin (π) \sin (- x) + \sin (\frac{π}{2}) \cos (x) + \cos (\frac{π}{2}) \sin (x)$

Step 4

Simplify the expression.

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Step 4.1

Simplify each term.

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Step 4.1.1

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.

$- \cos (0) \cos (- x) - \sin (π) \sin (- x) + \sin (\frac{π}{2}) \cos (x) + \cos (\frac{π}{2}) \sin (x)$

Step 4.1.2

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

$- 1 \cdot 1 \cos (- x) - \sin (π) \sin (- x) + \sin (\frac{π}{2}) \cos (x) + \cos (\frac{π}{2}) \sin (x)$

Step 4.1.3

Multiply $- 1$ by $1$ .

$- 1 \cos (- x) - \sin (π) \sin (- x) + \sin (\frac{π}{2}) \cos (x) + \cos (\frac{π}{2}) \sin (x)$

Step 4.1.4

Since $\cos (- x)$ is an even function, rewrite $\cos (- x)$ as $\cos (x)$ .

$- 1 \cos (x) - \sin (π) \sin (- x) + \sin (\frac{π}{2}) \cos (x) + \cos (\frac{π}{2}) \sin (x)$

Step 4.1.5

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

$- \cos (x) - \sin (π) \sin (- x) + \sin (\frac{π}{2}) \cos (x) + \cos (\frac{π}{2}) \sin (x)$

Step 4.1.6

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

$- \cos (x) - \sin (0) \sin (- x) + \sin (\frac{π}{2}) \cos (x) + \cos (\frac{π}{2}) \sin (x)$

Step 4.1.7

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

$- \cos (x) - 0 \sin (- x) + \sin (\frac{π}{2}) \cos (x) + \cos (\frac{π}{2}) \sin (x)$

Step 4.1.8

Multiply $- 1$ by $0$ .

$- \cos (x) + 0 \sin (- x) + \sin (\frac{π}{2}) \cos (x) + \cos (\frac{π}{2}) \sin (x)$

Step 4.1.9

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

$- \cos (x) + 0 (- \sin (x)) + \sin (\frac{π}{2}) \cos (x) + \cos (\frac{π}{2}) \sin (x)$

Step 4.1.10

Multiply $0 (- \sin (x))$ .

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Step 4.1.10.1

Multiply $- 1$ by $0$ .

$- \cos (x) + 0 \sin (x) + \sin (\frac{π}{2}) \cos (x) + \cos (\frac{π}{2}) \sin (x)$

Step 4.1.10.2

Multiply $0$ by $\sin (x)$ .

$- \cos (x) + 0 + \sin (\frac{π}{2}) \cos (x) + \cos (\frac{π}{2}) \sin (x)$

Step 4.1.11

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$- \cos (x) + 0 + 1 \cos (x) + \cos (\frac{π}{2}) \sin (x)$

Step 4.1.12

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

$- \cos (x) + 0 + \cos (x) + \cos (\frac{π}{2}) \sin (x)$

Step 4.1.13

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$- \cos (x) + 0 + \cos (x) + 0 \sin (x)$

Step 4.1.14

Multiply $0$ by $\sin (x)$ .

$- \cos (x) + 0 + \cos (x) + 0$

Step 4.2

Combine the opposite terms in $- \cos (x) + 0 + \cos (x) + 0$ .

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Step 4.2.1

Add $- \cos (x)$ and $0$ .

$- \cos (x) + \cos (x) + 0$

Step 4.2.2

Add $- \cos (x) + \cos (x)$ and $0$ .

$- \cos (x) + \cos (x)$

Step 4.2.3

Add $- \cos (x)$ and $\cos (x)$ .

$0$

Step 5

Because the two sides have been shown to be equivalent, the equation is an identity.

$\cos (π - x) + \sin (\frac{π}{2} + x) = 0$ is an identity