In Exercises 35–38, find the exact value of the following under the given conditions:
a. sin(α + β)
sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.

검증된 단계별 안내

Identify the given information: \( \sin \alpha = \frac{3}{5} \) with \( 0 < \alpha < \frac{\pi}{2} \), and \( \sin \beta = \frac{12}{13} \) with \( \frac{\pi}{2} < \beta < \pi \).

Determine the quadrants for \( \alpha \) and \( \beta \) to find the signs of \( \cos \alpha \) and \( \cos \beta \). Since \( \alpha \) is in the first quadrant, \( \cos \alpha > 0 \). Since \( \beta \) is in the second quadrant, \( \cos \beta < 0 \).

Use the Pythagorean identity to find \( \cos \alpha \) and \( \cos \beta \): \[ \cos \theta = \pm \sqrt{1 - \sin^2 \theta} \] Calculate: \[ \cos \alpha = +\sqrt{1 - \left(\frac{3}{5}\right)^2} \] \[ \cos \beta = -\sqrt{1 - \left(\frac{12}{13}\right)^2} \]

Apply the sine addition formula: \[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \]

Substitute the known values of \( \sin \alpha \), \( \cos \alpha \), \( \sin \beta \), and \( \cos \beta \) into the formula to express \( \sin(\alpha + \beta) \) exactly.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Sum of Angles Formula for Sine

The sum of angles formula states that sin(α + β) = sin α cos β + cos α sin β. This identity allows us to find the sine of a sum of two angles using the sine and cosine of each individual angle, which is essential for solving the problem.

Determining Cosine from Sine and Quadrant

Given sin θ and the quadrant of angle θ, we can find cos θ using the Pythagorean identity cos² θ = 1 - sin² θ. The sign of cos θ depends on the quadrant, so knowing the angle's range is crucial to assign the correct positive or negative value.

Understanding Angle Ranges and Quadrants

The given angle ranges (e.g., 0 < α < π/2) indicate the quadrant in which each angle lies. This information helps determine the signs of sine and cosine values, which is necessary for correctly applying trigonometric identities and finding exact values.

In Exercises 35–38, find the exact value of the following under the given conditions: a. sin(α + β)sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.