body{margin:0;} noscript{z-index: 10000; position: fixed; display: block; height: 100%; width: 100%; background: #fff;} noscript .fa{font-size: 40px; display:block; color: #daa520;} noscript div{margin-top: 6%; padding-left: 20px; padding-right: 20px; text-align: center;} ! You must have JavaScript enabled to use this site.

Big Ideas Math Algebra 2, 2014

BI

Big Ideas Math Algebra 2, 2014 View details

8. Using Sum and Difference Formulas

Continue to next subchapter

Exercise 10 Page 523

Before we rewrite the given expression, let's start by recalling the values of the three main trigonometric functions for the most important angles.

	$sin θ$	$cos θ$	$tan θ$
$θ = 0$	$0$	$1$	$0$
$θ = \frac{π}{6}$	$\frac{1}{2}$	$\frac{3}{2}$	$\frac{3}{3}$
$θ = \frac{π}{4}$	$\frac{2}{2}$	$\frac{2}{2}$	$1$
$θ = \frac{π}{3}$	$\frac{3}{2}$	$\frac{1}{2}$	$3$
$θ = \frac{π}{2}$	$1$	$0$	-
$θ = π$	$0$	$- 1$	$0$
$θ = 2 π$	$0$	$1$	$0$

Let's now recall the Angle Difference Formula for a sine.

sin (A - B) = sin A cos B - cos A sin B

We will use this identity to rewrite the given expression.

sin \frac{- 7 π}{1 2}

Write as a difference

sin (\frac{- π}{2} - \frac{π}{1 2})

$sin (A - B) = s i n A c o s B - c o s A s i n B$

sin \frac{- π}{2} cos \frac{π}{1 2} - cos \frac{- π}{2} sin \frac{π}{1 2}

Now, we will recall the Sine and Cosine Negative Angle Identities.

\underline{Sine Negative Angle Identity} sin (- θ) = - sin θ \underline{Cosine Negative Angle Identity} cos (- θ) = cos θ

Next, we will use these identities, to simplify this expression.

sin \frac{- π}{2} cos \frac{π}{1 2} - cos \frac{- π}{2} sin \frac{π}{1 2}

$cos (- θ) = cos (θ)$

sin \frac{- π}{2} cos \frac{π}{1 2} - cos \frac{π}{2} sin \frac{π}{1 2}

$sin (- θ) = - sin (θ)$

- sin \frac{π}{2} cos \frac{π}{1 2} - cos \frac{π}{2} sin \frac{π}{1 2}

Next, we will use the table we constructed at the beginning of this solution to simplify this expression.

- s i n \frac{π}{2} c o s \frac{π}{1 2} - c o s \frac{π}{2} s i n \frac{π}{1 2} = (- 1) c o s \frac{π}{1 2} - (0) s i n \frac{π}{1 2}

Let's simplify the obtained expression!

(- 1) c o s \frac{π}{1 2} - (0) s i n \frac{π}{1 2}

▼

Simplify

Zero Property of Multiplication

(- 1) cos \frac{π}{1 2} - 0

Identity Property of Multiplication

- cos \frac{π}{1 2} - 0

Identity Property of Addition

- cos \frac{π}{1 2}

Therefore, we found that

sin \frac{- 7 π}{1 2} = - cos \frac{π}{1 2} .

Be aware that

\frac{π}{1 2}

is the difference of

\frac{π}{3}

and

\frac{π}{4} .

Therefore, we can rewrite

cos \frac{π}{1 2}

as the cosine of a difference.

cos \frac{π}{1 2} = cos (\frac{π}{3} - \frac{π}{4})

Let's now recall the Angle Difference Formula for a cosine.

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

We can use this identity to rewrite the

cos \frac{π}{1 2} .

cos \frac{π}{1 2}

Write as a difference

cos (\frac{π}{3} - \frac{π}{4})

$cos (A - B) = c o s A c o s B + s i n A s i n B$

cos \frac{π}{3} cos \frac{π}{4} + sin \frac{π}{3} sin \frac{π}{4}

Next, we will use the table we constructed at the beginning of this solution to simplify this expression.

c o s \frac{π}{3} c o s \frac{π}{4} + s i n \frac{π}{3} s i n \frac{π}{4} = (\frac{1}{2}) \frac{2}{2} + (\frac{3}{2}) \frac{2}{2}

Let's finally simplify the obtained expression!

(\frac{1}{2}) \frac{2}{2} + (\frac{3}{2}) \frac{2}{2}

▼

Simplify

Multiply fractions

\frac{1 \cdot 2}{2 \cdot 2} + \frac{3 \cdot 2}{2 \cdot 2}

Identity Property of Multiplication

\frac{2}{2 \cdot 2} + \frac{3 \cdot 2}{2 \cdot 2}

$a \cdot b = a \cdot b$

\frac{2}{2 \cdot 2} + \frac{3 \cdot 2}{2 \cdot 2}

Multiply

\frac{2}{4} + \frac{6}{4}

Add fractions

\frac{2 + 6}{4}

Therefore,

cos \frac{π}{1 2} = \frac{2 + 6}{4} .

We can finally use this information to calculate our original expression.

sin \frac{- 7 π}{1 2} = - cos \frac{π}{1 2}

$cos \frac{π}{1 2} = \frac{2 + 6}{4}$

sin \frac{- 7 π}{1 2} = - \frac{2 + 6}{4}

Subchapter links

Communicate Your Answer p.519

Monitoring Progress p.521-522