Math  /  Trigonometry

QuestionUse an Addition or Subtraction Formula to write the expression as a trigonometric function of one number. cos(14)cos(16)sin(14)sin(16)\cos \left(14^{\circ}\right) \cos \left(16^{\circ}\right)-\sin \left(14^{\circ}\right) \sin \left(16^{\circ}\right) \square
Find its exact value. \square Need Help? \square Read It
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Studdy Solution

STEP 1

What is this asking? We're asked to rewrite the given expression using a single trigonometric function and then find its exact value. Watch out! Be careful with the signs in the addition and subtraction formulas!

STEP 2

1. Identify the relevant formula
2. Apply the formula
3. Calculate the result

STEP 3

Alright, let's look at this exciting expression: cos(14)cos(16)sin(14)sin(16)\cos(14^{\circ}) \cdot \cos(16^{\circ}) - \sin(14^{\circ}) \cdot \sin(16^{\circ}).
Does it remind you of anything?
It looks a lot like the cosine addition formula!

STEP 4

Remember, the **cosine addition formula** is cos(a+b)=cos(a)cos(b)sin(a)sin(b)\cos(a + b) = \cos(a) \cdot \cos(b) - \sin(a) \cdot \sin(b).
See the similarity?
Our expression matches this formula perfectly, with a=14a = 14^{\circ} and b=16b = 16^{\circ}.

STEP 5

Now, let's **substitute** our values into the formula.
We have a=14a = 14^{\circ} and b=16b = 16^{\circ}, so our expression becomes cos(14+16)\cos(14^{\circ} + 16^{\circ}).

STEP 6

This simplifies to cos(30)\cos(30^{\circ}).
Much nicer, right?
We've rewritten the original expression as a trigonometric function of a single number!

STEP 7

Now, we just need to find the **exact value** of cos(30)\cos(30^{\circ}).
Remember your **special triangles**!

STEP 8

The cosine of 3030^{\circ} is 32\frac{\sqrt{3}}{2}.
So, cos(30)=32\cos(30^{\circ}) = \frac{\sqrt{3}}{2}.

STEP 9

The expression can be rewritten as cos(30)\cos(30^{\circ}), and its exact value is 32\frac{\sqrt{3}}{2}.

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