Questions: Express cos Q as a fraction in simplest terms.

Express cos Q as a fraction in simplest terms.
Transcript text: Express $\cos Q$ as a fraction in simplest terms.
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Solution

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Solution Steps

Step 1: Identify the sides of the triangle relative to angle Q.

Side SQ is adjacent to angle Q, side RQ is opposite to angle Q, and side QS is the hypotenuse. The lengths of the adjacent side and the hypotenuse are given as 12 and $\sqrt{95}$ respectively.

Step 2: Recall the definition of cosine.

The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

Step 3: Calculate $\cos Q$.

$\cos Q = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{SQ}{RQ} = \frac{12}{\sqrt{95}}$

Step 4: Rationalize the denominator.

Multiply the numerator and denominator by $\sqrt{95}$ to get rid of the square root in the denominator.

$\cos Q = \frac{12}{\sqrt{95}} \times \frac{\sqrt{95}}{\sqrt{95}} = \frac{12\sqrt{95}}{95}$

Final Answer The final answer is $\boxed{\frac{12\sqrt{95}}{95}}$

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