in $$ 28^{\circ}=a $$ and $$ \cos 32^{\circ}=b $$, determine the following in terms of a and/or $$ 1 \cos 28^{\circ} $$ $$ 2 \cos 64^{*} $$

Question

UpStudy AI · Accepted Answer

Let's clarify the given information and address each part step by step.

### Given:
- Let $$ a = \cos 28^\circ $$
- Let $$ b = \cos 32^\circ $$

### Objectives:
1. Express $$ \cos 28^\circ $$ in terms of $$ a $$ and/or $$ b $$.
2. Express $$ \cos 64^\circ $$ in terms of $$ a $$ and/or $$ b $$.

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### 1. Expressing $$ \cos 28^\circ $$:

Since $$ a = \cos 28^\circ $$, this part is straightforward.

$$
\cos 28^\circ = a
$$

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### 2. Expressing $$ \cos 64^\circ $$:

To express $$ \cos 64^\circ $$ in terms of $$ b $$, we can utilize the **Double Angle Identity** for cosine:

$$
\cos 2\theta = 2\cos^2 \theta - 1
$$

Here, we can let $$ \theta = 32^\circ $$, which gives:

$$
\cos 64^\circ = 2\cos^2 32^\circ - 1
$$

Since $$ b = \cos 32^\circ $$, substitute $$ b $$ into the equation:

$$
\cos 64^\circ = 2b^2 - 1
$$

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### **Final Answers:**

1. $$ \cos 28^\circ = a $$

2. $$ \cos 64^\circ = 2b^2 - 1 $$
$$ \cos 28^\circ = a $$ $$ \cos 64^\circ = 2b^2 - 1 $$

in \( 28^{\circ}=a \) and \( \cos 32^{\circ}=b \), determine the following in terms of a and/or \( 1 \cos 28^{\circ} \) \( 2 \cos 64^{*} \)