Concept Check Suppose that sec θ = (x+4)/x.
Find an expression in x for tan θ.

검증된 단계별 안내

Recall the trigonometric identity: \( \sec \theta = \frac{1}{\cos \theta} \).

Given \( \sec \theta = \frac{x+4}{x} \), we can express \( \cos \theta \) as \( \cos \theta = \frac{x}{x+4} \).

Use the Pythagorean identity: \( \tan^2 \theta = \sec^2 \theta - 1 \).

Substitute \( \sec \theta = \frac{x+4}{x} \) into the identity to find \( \tan^2 \theta = \left(\frac{x+4}{x}\right)^2 - 1 \).

Simplify the expression for \( \tan^2 \theta \) and then take the square root to find \( \tan \theta \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

도움이 되었나요?

주요 개념

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

Secant Function

The secant function, denoted as sec θ, is the reciprocal of the cosine function. It is defined as sec θ = 1/cos θ. In this context, sec θ = (x+4)/x implies a relationship between the angle θ and the variable x, which can be used to derive other trigonometric functions.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, the relationship sin² θ + cos² θ = 1 holds true. This identity can be rearranged to express tan θ in terms of sec θ, as tan² θ = sec² θ - 1. Understanding this identity is crucial for deriving tan θ from sec θ.

Tangent Function

The tangent function, denoted as tan θ, is defined as the ratio of the sine and cosine functions: tan θ = sin θ/cos θ. It can also be expressed in terms of secant as tan θ = √(sec² θ - 1). This relationship allows us to find an expression for tan θ using the given sec θ value.

Concept Check Suppose that sec θ = (x+4)/x.Find an expression in x for tan θ.