Using the Half-Angle Formulas

Find the function values in Exercises 47–50.

sin² 3π/8

검증된 단계별 안내

First, recognize that the half-angle formulas are useful for finding the sine or cosine of an angle that is half of a given angle. The half-angle formula for sine is: \( \sin^2 \theta = \frac{1 - \cos(2\theta)}{2} \).

In this problem, we need to find \( \sin^2 \frac{3\pi}{8} \). Notice that \( \frac{3\pi}{8} \) is half of \( \frac{3\pi}{4} \). Therefore, we can use the half-angle formula for sine.

Substitute \( \theta = \frac{3\pi}{8} \) into the half-angle formula: \( \sin^2 \frac{3\pi}{8} = \frac{1 - \cos(\frac{3\pi}{4})}{2} \).

Next, find \( \cos(\frac{3\pi}{4}) \). The angle \( \frac{3\pi}{4} \) is in the second quadrant where cosine is negative. The reference angle is \( \frac{\pi}{4} \), and \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \). Therefore, \( \cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} \).

Substitute \( \cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} \) back into the formula: \( \sin^2 \frac{3\pi}{8} = \frac{1 - (-\frac{\sqrt{2}}{2})}{2} \). Simplify the expression to find the value of \( \sin^2 \frac{3\pi}{8} \).

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주요 개념

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Half-Angle Formulas

Half-angle formulas are trigonometric identities that express the sine and cosine of half an angle in terms of the sine and cosine of the original angle. For example, the sine half-angle formula is given by sin(θ/2) = √((1 - cos(θ))/2). These formulas are particularly useful for simplifying expressions involving angles that are not easily computable.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. They are fundamental in calculus and are used to model periodic phenomena. Understanding how to evaluate these functions at specific angles, including those expressed in radians, is essential for solving problems involving trigonometric identities.

Radians and Degrees

Radians and degrees are two units for measuring angles. Radians are based on the radius of a circle, where one radian is the angle subtended by an arc equal in length to the radius. Understanding the conversion between these units is crucial, especially when working with trigonometric functions, as many calculus problems use radians for angle measurements.

Using the Half-Angle FormulasFind the function values in Exercises 47–50.sin² 3π/8