Math Problem Statement

Find the exact value of the expression.

cosine left parenthesis 60 degrees plus 225 degrees right parenthesiscos(60°+225°)

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Part 1

Apply the sum and difference identities. Choose the correct answer below.

A.cosine 60 degrees cosine 225 degrees plus sine 60 degrees sine 225 degrees

cosine 60 degrees cosine 225 degreescos60°cos225°plus+sine 60 degrees sine 225 degreessin60°sin225°

B.sine 60 degrees cosine 225 degrees minus cosine 60 degrees sine 225 degrees

sine 60 degrees cosine 225 degreessin60°cos225°minus−cosine 60 degrees sine 225 degreescos60°sin225°

C.StartFraction tangent 60 degrees minus tangent 225 degrees Over 1 plus tangent 60 degrees tangent 225 degrees EndFraction

StartFraction tangent 60 degrees minus tangent 225 degrees Over 1 plus tangent 60 degrees tangent 225 degrees EndFractiontan60°−tan225°1+tan60°tan225°

D.sine 60 degrees cosine 225 degrees plus cosine 60 degrees sine 225 degrees

sine 60 degrees cosine 225 degreessin60°cos225°plus+cosine 60 degrees sine 225 degreescos60°sin225°

E.cosine 225 degrees cosine 225 degrees plus sine 60 degrees sine 60 degrees

cosine 225 degrees cosine 225 degreescos225°cos225°plus+sine 60 degrees sine 60 degreessin60°sin60°

F.cosine 60 degrees cosine 60 degrees minus sine 225 degrees sine 225 degrees

cosine 60 degrees cosine 60 degreescos60°cos60°minus−sine 225 degrees sine 225 degreessin225°sin225°

G.cosine 60 degrees cosine 60 degrees plus sine 225 degrees sine 225 degrees

cosine 60 degrees cosine 60 degreescos60°cos60°plus+sine 225 degrees sine 225 degreessin225°sin225°

H.sine 60 degrees cosine 60 degrees minus cosine 225 degrees sine 225 degrees

sine 60 degrees cosine 60 degreessin60°cos60°minus−cosine 225 degrees sine 225 degreescos225°sin225°

I.StartFraction tangent 225 degrees minus tangent 60 degrees Over 1 minus tangent 60 degrees tangent 225 degrees EndFraction

StartFraction tangent 225 degrees minus tangent 60 degrees Over 1 minus tangent 60 degrees tangent 225 degrees EndFractiontan225°−tan60°1−tan60°tan225°

J.sine 60 degrees cosine 60 degrees plus cosine 225 degrees sine 225 degrees

sine 60 degrees cosine 60 degreessin60°cos60°plus+cosine 225 degrees sine 225 degreescos225°sin225°

K.sine 225 degrees cosine 225 degrees minus sine 60 degrees cosine 60 degrees

sine 225 degrees cosine 225 degreessin225°cos225°minus−sine 60 degrees cosine 60 degreessin60°cos60°

L.cosine 225 degrees cosine 225 degrees minus sine 60 degrees sine 60 degrees

cosine 225 degrees cosine 225 degreescos225°cos225°minus−sine 60 degrees sine 60 degreessin60°sin60°

M.StartFraction tangent 60 degrees plus tangent 225 degrees Over 1 minus tangent 60 degrees tangent 225 degrees EndFraction

StartFraction tangent 60 degrees plus tangent 225 degrees Over 1 minus tangent 60 degrees tangent 225 degrees EndFractiontan60°+tan225°1−tan60°tan225°

N.cosine 60 degrees cosine 225 degrees minus sine 60 degrees sine 225 degrees

cosine 60 degrees cosine 225 degreescos60°cos225°

Solution

The given problem involves applying the sum identity for cosine to compute cos⁡(60∘+225∘)\cos(60^\circ + 225^\circ). The sum identity for cosine is:

cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A + B) = \cos A \cos B - \sin A \sin B

Here, A=60∘A = 60^\circ and B=225∘B = 225^\circ. Using the formula:

cos⁡(60∘+225∘)=cos⁡60∘cos⁡225∘−sin⁡60∘sin⁡225∘\cos(60^\circ + 225^\circ) = \cos 60^\circ \cos 225^\circ - \sin 60^\circ \sin 225^\circ

Correct Answer:

The correct option is: N. cos⁡60∘cos⁡225∘−sin⁡60∘sin⁡225∘\cos 60^\circ \cos 225^\circ - \sin 60^\circ \sin 225^\circ

Would you like to evaluate this further and find the exact value, or review related concepts?

5 Related Questions:

  1. What are the exact values of cos⁡60∘\cos 60^\circ, sin⁡60∘\sin 60^\circ, cos⁡225∘\cos 225^\circ, and sin⁡225∘\sin 225^\circ?
  2. How does the cosine identity differ from the sine identity for sums?
  3. What quadrant does the angle 285∘285^\circ (i.e., 60∘+225∘60^\circ + 225^\circ) lie in, and what does that imply about its cosine?
  4. Can the sum identity for cosine be applied to angles in radians?
  5. How would you derive the tangent sum identity starting from sine and cosine sum identities?

Tip:

For trigonometric problems, always simplify using identities and verify the quadrant to determine the sign of the trigonometric functions.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Cosine Function
Sum of Angles Identity

Formulas

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

Theorems

Sum Identity for Cosine

Suitable Grade Level

Grades 9-12

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