In Exercises 9–16, use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.
sin 𝜋/4 - cos 𝜋/4

Verified step by step guidance

1. Identify the triangle as a 45°-45°-90° triangle, where the legs are equal and the hypotenuse is \( \sqrt{2} \).
2. Recall that for a 45°-45°-90° triangle, \( \sin 45° = \cos 45° = \frac{1}{\sqrt{2}} \).
3. Express \( \sin \frac{\pi}{4} \) and \( \cos \frac{\pi}{4} \) in terms of the triangle: \( \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \) and \( \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}} \).
4. Substitute these values into the expression \( \sin \frac{\pi}{4} - \cos \frac{\pi}{4} \).
5. Simplify the expression: \( \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} = 0 \).

Key Concepts

Trigonometric Ratios

Trigonometric ratios relate the angles of a triangle to the lengths of its sides. For a right triangle, the sine and cosine functions are defined as the ratios of the lengths of the opposite side to the hypotenuse and the adjacent side to the hypotenuse, respectively. In this case, for a 45° angle, both sine and cosine yield the same value, specifically √2/2, due to the isosceles nature of the triangle.

Rationalizing the Denominator

Rationalizing the denominator is a mathematical technique used to eliminate square roots or irrational numbers from the denominator of a fraction. This is achieved by multiplying both the numerator and the denominator by a suitable value that will result in a rational number in the denominator. This process is important for simplifying expressions and making them easier to interpret.

Special Angles in Trigonometry

Certain angles, such as 30°, 45°, and 60°, have known sine and cosine values that are commonly used in trigonometry. For example, sin(π/4) and cos(π/4) both equal √2/2. Recognizing these special angles allows for quicker calculations and a deeper understanding of trigonometric functions, especially when evaluating expressions involving these angles.