Prove that cos 2A = cos² A - sin² A. The diagram shows part of the circular end of a running track with three running lanes shown. The centre of each of the circular boundaries of the lanes is at O. Kate runs in the middle of lane 1, from A to B as shown. Helen runs in the middle of lane 2, from C to D as shown. Helen runs 3 m further than Kate. |∠AOB| = |∠COD| = θ radians. If each lane is 1-2 m wide, find θ.

Leaving Cert Mathematics Trigonometry: Prove that cos 2A = cos² A

Prove that cos 2A = cos² A - sin² A - Leaving Cert Mathematics - Question 2 - 2014

Question 2

Prove that cos 2A = cos² A - sin² A. The diagram shows part of the circular end of a running track with three running lanes shown. The centre of each of the circula... show full transcript

Worked Solution & Example Answer:Prove that cos 2A = cos² A - sin² A - Leaving Cert Mathematics - Question 2 - 2014

Step 1

Prove that cos 2A = cos² A - sin² A

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Answer

To prove that

ext{cos}(2A) = ext{cos}^2(A) - ext{sin}^2(A)

we can use the double angle formula for cosine. The double angle formula states:

ext{cos}(2A) = ext{cos}(A + A) = ext{cos}(A) ext{cos}(A) - ext{sin}(A) ext{sin}(A)

which simplifies to:

ext{cos}(2A) = ext{cos}^2(A) - ext{sin}^2(A).

Thus, the identity is verified.

Step 2

|AOB| = |COD| = θ radians

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Answer

We need to find θ based on the distances traveled by Kate and Helen. Let's denote:

Kate's distance: |AB| = $s_1 = r_0 θ$
Helen's distance: |CD| = $s_2 = r_1 θ + 1 - 2θ$ (since she runs 3 m further than Kate)

From the question, we know:

$s_1 + 3 = s_2$ Substituting the expressions for $s_1$ and $s_2$ gives us:

ext{r}_0 θ + 3 = (r_0 + 1)θ + (1 + 2θ).

Solving for θ yields:

1 - 2θ = r_0θ + 3 \ ightarrow r_0 + 3 = r_0θ + 1 - 2θ \ ightarrow 1 = θ + 2.5θ \ ightarrow θ = 2 - 5 ext{ radians}.

Prove that cos 2A = cos² A - sin² A - Leaving Cert Mathematics - Question 2 - 2014

Worked Solution & Example Answer:Prove that cos 2A = cos² A - sin² A - Leaving Cert Mathematics - Question 2 - 2014

Prove that cos 2A = cos² A - sin² A

|AOB| = |COD| = θ radians

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