In Exercises 57–64, find the exact value of the following under the given conditions:
c. tan (α + β)
cos α = 8/17, α lies in quadrant IV, and sin β = -1/2, β lies in quadrant III.

검증된 단계별 안내

Identify the given information: \(\cos \alpha = \frac{8}{17}\) with \(\alpha\) in quadrant IV, and \(\sin \beta = -\frac{1}{2}\) with \(\beta\) in quadrant III.

Determine the signs and values of \(\sin \alpha\) and \(\cos \beta\) using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), considering the quadrant of each angle.

Calculate \(\sin \alpha\) by using \(\sin \alpha = -\sqrt{1 - \cos^2 \alpha}\) since \(\alpha\) is in quadrant IV where sine is negative.

Calculate \(\cos \beta\) by using \(\cos \beta = -\sqrt{1 - \sin^2 \beta}\) since \(\beta\) is in quadrant III where cosine is negative.

Use the angle addition formula for tangent: \(\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\), where \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\) and \(\tan \beta = \frac{\sin \beta}{\cos \beta}\). Substitute the values found to express \(\tan(\alpha + \beta)\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

15m

도움이 되었나요?

주요 개념

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

Trigonometric Ratios and Quadrants

Trigonometric ratios (sine, cosine, tangent) relate the angles of a triangle to the ratios of its sides. The sign of these ratios depends on the quadrant in which the angle lies. For example, in quadrant IV, cosine is positive and sine is negative, while in quadrant III, both sine and cosine are negative.

Sum of Angles Formula for Tangent

The tangent of the sum of two angles α and β is given by tan(α + β) = (tan α + tan β) / (1 - tan α tan β). This formula allows us to find the exact value of tan(α + β) using the individual tangents of α and β, which can be derived from their sine and cosine values.

Finding Missing Trigonometric Values Using Pythagorean Identity

Given one trigonometric ratio and the quadrant, the other ratios can be found using the Pythagorean identity sin²θ + cos²θ = 1. For example, if cos α is known, sin α can be found by sin α = ±√(1 - cos²α), with the sign determined by the quadrant. This step is essential to compute tan α and tan β.

In Exercises 57–64, find the exact value of the following under the given conditions: c. tan (α + β) cos α = 8/17, α lies in quadrant IV, and sin β = -1/2, β lies in quadrant III.