Trigonometry Examples

sin (135)

$\sin (135)$

Step 1

The angle

135

$135$ is an angle where the values of the six trigonometric functions are known. Because this is the case, add

0

$0$ to keep the value the same.

sin (135 + 0)

$\sin (135 + 0)$

Step 2

Use the sum formula for sine to simplify the expression. The formula states that

sin (A + B) = sin (A) cos (B) + cos (A) sin (B)

$\sin (A + B) = \sin (A) \cos (B) + \cos (A) \sin (B)$ .

sin (135) cos (0) + cos (135) sin (0)

$\sin (135) \cos (0) + \cos (135) \sin (0)$

Step 3

Remove parentheses.

sin (135) cos (0) + cos (135) sin (0)

$\sin (135) \cos (0) + \cos (135) \sin (0)$

Step 4

Simplify each term.

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Step 4.1

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

sin (45) cos (0) + cos (135) sin (0)

$\sin (45) \cos (0) + \cos (135) \sin (0)$

Step 4.2

The exact value of

sin (45)

$\sin (45)$ is

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

\frac{\sqrt{2}}{2} cos (0) + cos (135) sin (0)

$\frac{\sqrt{2}}{2} \cos (0) + \cos (135) \sin (0)$

Step 4.3

The exact value of

cos (0)

$\cos (0)$ is

1

$1$ .

\frac{\sqrt{2}}{2} \cdot 1 + cos (135) sin (0)

$\frac{\sqrt{2}}{2} \cdot 1 + \cos (135) \sin (0)$

Step 4.4

Multiply

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ by

1

$1$ .

\frac{\sqrt{2}}{2} + cos (135) sin (0)

$\frac{\sqrt{2}}{2} + \cos (135) \sin (0)$

Step 4.5

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{\sqrt{2}}{2} - cos (45) sin (0)

$\frac{\sqrt{2}}{2} - \cos (45) \sin (0)$

Step 4.6

The exact value of

cos (45)

$\cos (45)$ is

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} sin (0)

$\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \sin (0)$

Step 4.7

The exact value of

sin (0)

$\sin (0)$ is

0

$0$ .

\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \cdot 0

$\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \cdot 0$

Step 4.8

Multiply

- \frac{\sqrt{2}}{2} \cdot 0

$- \frac{\sqrt{2}}{2} \cdot 0$ .

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Step 4.8.1

Multiply

0

$0$ by

- 1

$- 1$ .

\frac{\sqrt{2}}{2} + 0 \frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2} + 0 \frac{\sqrt{2}}{2}$

Step 4.8.2

Multiply

0

$0$ by

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

\frac{\sqrt{2}}{2} + 0

$\frac{\sqrt{2}}{2} + 0$

\frac{\sqrt{2}}{2} + 0

$\frac{\sqrt{2}}{2} + 0$

\frac{\sqrt{2}}{2} + 0

$\frac{\sqrt{2}}{2} + 0$

Step 5

Add

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ and

0

$0$ .

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$

Decimal Form:

0.70710678 \dots

$0.70710678 \dots$

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