Math / Trigonometry

QuestionUse the sum-to-product formulas to find the exact value of the expression. $9 \cos 120^{\circ}+9 \cos 60^{\circ}$
Step 1 Factor out the constant. $9 \cos 120^{\circ}+9 \cos 60^{\circ}=$ $\square$ $\left(\cos 120^{\circ}+\cos 60^{\circ}\right)$

Studdy Solution

STEP 1

1. The expression involves trigonometric functions that can be simplified using sum-to-product identities.
2. The sum-to-product identities are applicable to cosine functions.
3. We need to find the exact value of the expression.

STEP 2

1. Factor out the constant from the expression.
2. Apply the sum-to-product formula for cosine.
3. Simplify the resulting expression to find the exact value.

STEP 3

Factor out the constant from the expression:
$9 \cos 120^{\circ} + 9 \cos 60^{\circ} = 9 (\cos 120^{\circ} + \cos 60^{\circ})$

STEP 4

Apply the sum-to-product formula for cosine:
The sum-to-product formula for cosine is:
$\cos A + \cos B = 2 \cos \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right)$
For $A = 120^{\circ}$ and $B = 60^{\circ}$ :
$\cos 120^{\circ} + \cos 60^{\circ} = 2 \cos \left( \frac{120^{\circ} + 60^{\circ}}{2} \right) \cos \left( \frac{120^{\circ} - 60^{\circ}}{2} \right)$
$= 2 \cos (90^{\circ}) \cos (30^{\circ})$

STEP 5

Simplify the expression using known cosine values:
$\cos (90^{\circ}) = 0$
Thus,
$2 \cos (90^{\circ}) \cos (30^{\circ}) = 2 \times 0 \times \cos (30^{\circ}) = 0$
Therefore, the expression becomes:
$9 \times 0 = 0$
The exact value of the expression is:
$\boxed{0}$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms