Trigonometry Examples

$\cos (\frac{π}{5} + \frac{π}{6})$

Step 1

Use the sum formula for cosine to simplify the expression. The formula states that $\cos (A + B) = - (\cos (A) \cos (B) + \sin (A) \sin (B))$ .

$\cos (\frac{π}{6}) \cdot \cos (\frac{π}{5}) - \sin (\frac{π}{6}) \cdot \sin (\frac{π}{5})$

Step 2

Remove parentheses.

$\cos (\frac{π}{6}) \cdot \cos (\frac{π}{5}) - \sin (\frac{π}{6}) \cdot \sin (\frac{π}{5})$

Step 3

Simplify each term.

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Step 3.1

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{3}}{2} \cdot \cos (\frac{π}{5}) - \sin (\frac{π}{6}) \cdot \sin (\frac{π}{5})$

Step 3.2

Evaluate $\cos (\frac{π}{5})$ .

$\frac{\sqrt{3}}{2} \cdot 0.80901699 - \sin (\frac{π}{6}) \cdot \sin (\frac{π}{5})$

Step 3.3

Multiply $\frac{\sqrt{3}}{2} \cdot 0.80901699$ .

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Step 3.3.1

Combine $\frac{\sqrt{3}}{2}$ and $0.80901699$ .

$\frac{\sqrt{3} \cdot 0.80901699}{2} - \sin (\frac{π}{6}) \cdot \sin (\frac{π}{5})$

Step 3.3.2

Multiply $\sqrt{3}$ by $0.80901699$ .

$\frac{1.40125853}{2} - \sin (\frac{π}{6}) \cdot \sin (\frac{π}{5})$

Step 3.4

Divide $1.40125853$ by $2$ .

$0.70062926 - \sin (\frac{π}{6}) \cdot \sin (\frac{π}{5})$

Step 3.5

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$0.70062926 - \frac{1}{2} \cdot \sin (\frac{π}{5})$

Step 3.6

Evaluate $\sin (\frac{π}{5})$ .

$0.70062926 - \frac{1}{2} \cdot 0.58778525$

Step 3.7

Multiply $- \frac{1}{2} \cdot 0.58778525$ .

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Step 3.7.1

Multiply $0.58778525$ by $- 1$ .

$0.70062926 - 0.58778525 (\frac{1}{2})$

Step 3.7.2

Combine $- 0.58778525$ and $\frac{1}{2}$ .

$0.70062926 + \frac{- 0.58778525}{2}$

Step 3.8

Divide $- 0.58778525$ by $2$ .

$0.70062926 - 0.29389262$

Step 4

Subtract $0.29389262$ from $0.70062926$ .

$0.40673664$