Question

find the exact value of cos 45°.
cos 45° =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall the 45-45-90 triangle properties

In a 45-45-90 right triangle, the two legs are equal, and the hypotenuse \( c \) is related to the leg length \( a \) by \( c = a\sqrt{2} \) (from the Pythagorean theorem \( c^{2}=a^{2} + a^{2}=2a^{2}\), so \( c = a\sqrt{2} \)).

Step2: Define cosine in a right triangle

The cosine of an angle \( \theta \) in a right triangle is defined as \( \cos\theta=\frac{\text{adjacent side}}{\text{hypotenuse}} \). For a \( 45^{\circ} \) angle, the adjacent side and the opposite side (since it's a 45-45-90 triangle) are equal (let's say length \( a \)), and the hypotenuse is \( a\sqrt{2} \).

Step3: Calculate \( \cos45^{\circ} \)

Substitute into the cosine formula: \( \cos45^{\circ}=\frac{a}{a\sqrt{2}} \). The \( a \) terms cancel out (assuming \( a
eq0 \)), so we get \( \frac{1}{\sqrt{2}} \). Rationalizing the denominator (multiplying numerator and denominator by \( \sqrt{2} \)) gives \( \frac{\sqrt{2}}{2} \).

Answer:

\( \frac{\sqrt{2}}{2} \)