Find the exact value of cos 45°.
Answer (1)
Recall the definition of cosine in a right triangle as the ratio of the adjacent side to the hypotenuse.
Consider a 45-45-90 triangle with legs of length 1, and calculate the hypotenuse using the Pythagorean theorem: 2 .
Determine cos 45° as 2 1 and rationalize the denominator to get 2 2 .
State the exact value of cos 45°: 2 2 .
Explanation
Problem Analysis The problem asks us to find the exact value of cos 45 degrees. We are given that cos 45° = 0, which is incorrect. We need to find the correct value.
Using a 45-45-90 Triangle To find the exact value of cos 45°, we can use the unit circle or a 45-45-90 triangle. Let's consider a 45-45-90 triangle. In such a triangle, the two legs are equal in length, and the angles are 45°, 45°, and 90°. Let's assume the length of each leg is 1.
Finding the Hypotenuse Using the Pythagorean theorem, we can find the length of the hypotenuse. If a and b are the lengths of the legs and c is the length of the hypotenuse, then a 2 + b 2 = c 2 . In our case, 1 2 + 1 2 = c 2 , so 1 + 1 = c 2 , which means c 2 = 2 . Therefore, c = 2 $
Calculating Cosine Now, we can use the definition of cosine in a right triangle: cos(angle) = adjacent / hypotenuse. For a 45° angle in our triangle, the adjacent side has length 1, and the hypotenuse has length 2 . Therefore, cos 45° = 2 1
Rationalizing the Denominator To rationalize the denominator, we multiply the numerator and denominator by 2 : 2 1 × 2 2 = 2 2 So, the exact value of cos 45° is 2 2 .
Final Answer Therefore, the exact value of cos 45° is 2 2 .
Examples
Understanding the cosine of 45 degrees is crucial in various fields. For instance, in construction, when building a ramp at a 45-degree angle, knowing that cos 45° = 2 2 helps calculate the horizontal distance covered for a given ramp length. Similarly, in physics, when analyzing projectile motion at a 45-degree launch angle, this value is essential for determining the range of the projectile. This concept also applies to fields like photography and computer graphics, where angles and distances need to be precisely calculated.
