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

검증된 단계별 안내

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.

Recall the definitions of sine and cosine for angle \(\pi/4\) (which is 45°): \(\sin(\pi/4) = \frac{\text{opposite}}{\text{hypotenuse}}\) and \(\cos(\pi/4) = \frac{\text{adjacent}}{\text{hypotenuse}}\).

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 \(\sqrt{2}\).

Set up the expression \(\sin(\pi/4) - \cos(\pi/4)\) and substitute the values found: \(\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}\).

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.

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주요 개념

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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.

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