In Exercises 57–64, find the exact value of the following under the given conditions: b. sin (α + β), sin α = 5/6 , 𝝅/2 < α < 𝝅 , and tan β = 3/7 , 𝝅 < β < 3𝝅/2 .

검증된 단계별 안내

Identify the given information: \( \sin \alpha = \frac{5\pi}{6} \) with \( \frac{3\pi}{2} < \alpha < \pi \), and \( \tan \beta = \frac{3\pi}{2} \) with \( \frac{7\pi}{2} < \beta < \text{(missing upper bound)} \). Note that the interval for \( \alpha \) seems inconsistent since \( \frac{3\pi}{2} > \pi \). Verify the intervals and values carefully before proceeding.

Recall the formula for \( \sin(\alpha + \beta) \): \[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \]

Since \( \sin \alpha \) is given, find \( \cos \alpha \) using the Pythagorean identity: \[ \cos \alpha = \pm \sqrt{1 - \sin^2 \alpha} \] Determine the correct sign of \( \cos \alpha \) based on the quadrant where \( \alpha \) lies.

Use the given \( \tan \beta \) to find \( \sin \beta \) and \( \cos \beta \). Recall that: \[ \tan \beta = \frac{\sin \beta}{\cos \beta} \] Use the Pythagorean identity to express \( \sin \beta \) and \( \cos \beta \) in terms of \( \tan \beta \), and determine their signs based on the quadrant of \( \beta \).

Substitute the values of \( \sin \alpha \), \( \cos \alpha \), \( \sin \beta \), and \( \cos \beta \) into the formula for \( \sin(\alpha + \beta) \) and simplify to find the exact value.

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 sines and cosines of the individual angles, which is essential when given trigonometric values of α and β separately.

Determining the Sign of Trigonometric Functions Based on Quadrants

The signs of sine, cosine, and tangent depend on the quadrant in which the angle lies. Knowing the interval for α and β helps determine whether sine, cosine, or tangent values are positive or negative, which is crucial for correctly evaluating trigonometric expressions.

Using Given Trigonometric Ratios to Find Missing Values

Given sin α and tan β, we can use Pythagorean identities to find cos α and cos β. For example, cos α = ±√(1 - sin² α), with the sign determined by the quadrant. Similarly, from tan β, we find sin β and cos β using the identity tan β = sin β / cos β, enabling full evaluation of sin(α + β).