Solve the equation 8 cos2
1 − 10 cos 1 + 2 = 0 for 0Å ≤ 1 ≤ 180Å.

To solve the equation \(8 \cos^2 \theta - 10 \cos \theta + 2 = 0\) for \(0^\circ \leq \theta \leq 180^\circ\), we can treat it as a quadratic equation in terms of \(\cos \theta\).

### Step 1: Identify the quadratic equation
The equation is:

\[
8 \cos^2 \theta - 10 \cos \theta + 2 = 0
\]

Let \( x = \cos \theta \). Substituting this into the equation gives:

\[
8x^2 - 10x + 2 = 0
\]

### Step 2: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \(a = 8\), \(b = -10\), and \(c = 2\). Substituting these values into the formula:

\[
x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(8)(2)}}{2(8)}
\]

\[
x = \frac{10 \pm \sqrt{100 - 64}}{16}
\]

\[
x = \frac{10 \pm \sqrt{36}}{16}
\]

\[
x = \frac{10 \pm 6}{16}
\]

This gives us two possible values for \(x\):

\[
x = \frac{10 + 6}{16} = \frac{16}{16} = 1
\]

\[
x = \frac{10 - 6}{16} = \frac{4}{16} = \frac{1}{4}
\]

### Step 3: Find the corresponding values of \( \theta \)
Now, we need to find the angles \( \theta \) such that:

1. \( \cos \theta = 1 \)
2. \( \cos \theta = \frac{1}{4} \)

For \( \cos \theta = 1 \):

\[
\theta = 0^\circ
\]

For \( \cos \theta = \frac{1}{4} \):

To find \( \theta \) for \( \cos \theta = \frac{1}{4} \), we use the inverse cosine function:

\

Learn how to solve the trigonometric equation 8 cos^2 θ - 10 cos θ + 2 = 0 step-by-step. This problem involves applying trigonometric identities and solving a quadratic equation in terms of cos θ. Suitable for high school students studying trigonometry.

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