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 given triangle as a 45°-45°-90° right triangle, where the legs are equal and the hypotenuse is \(\sqrt{2}\) times the length of each leg.
2. Recall the definitions of sine and cosine for angle \(\pi/4\) (which is 45°): \(\sin(\pi/4) = \frac{\text{opposite}}{\text{hypotenuse}}\) and \(\cos(\pi/4) = \frac{\text{adjacent}}{\text{hypotenuse}}\).
3. Using the triangle, find \(\sin(\pi/4)\) as \(\frac{1}{\sqrt{2}}\) and \(\cos(\pi/4)\) as \(\frac{1}{\sqrt{2}}\) because both legs are 1 and the hypotenuse is \(\sqrt{2}\).
4. Set up the expression \(\sin(\pi/4) - \cos(\pi/4)\) and substitute the values found: \(\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}\).
5. Simplify the expression and if necessary, rationalize the denominator by multiplying numerator and denominator by \(\sqrt{2}\) to eliminate the square root in the denominator.

Key Concepts

45°-45°-90° Triangle Properties

A 45°-45°-90° triangle is an isosceles right triangle where the legs are congruent, and the hypotenuse is √2 times the length of each leg. This relationship helps in determining side lengths and trigonometric ratios for angles of 45°.

Trigonometric Ratios for 45°

For a 45° angle in a right triangle, both sine and cosine values are equal because the legs opposite and adjacent to the angle are the same length. Specifically, sin(45°) = cos(45°) = √2/2, which simplifies calculations involving these angles.

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any square roots from the denominator of a fraction by multiplying numerator and denominator by a suitable radical. This process simplifies expressions and is often required for final answers in trigonometry problems.