In Exercises 29–51, find the exact value of each expression. Do not use a calculator. _ csc(tan⁻¹ √3/3)

검증된 단계별 안내

Recognize that the expression is \( \csc(\tan^{-1}(\frac{\sqrt{3}}{3})) \). The first step is to let \( \theta = \tan^{-1}(\frac{\sqrt{3}}{3}) \), which means \( \tan(\theta) = \frac{\sqrt{3}}{3} \).

Recall that \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). Assign the opposite side as \( \sqrt{3} \) and the adjacent side as 3 to form a right triangle representing \( \theta \).

Use the Pythagorean theorem to find the hypotenuse \( h \) of the triangle: \( h = \sqrt{(\sqrt{3})^2 + 3^2} = \sqrt{3 + 9} = \sqrt{12} \). Simplify \( \sqrt{12} \) to \( 2\sqrt{3} \).

Since \( \csc(\theta) = \frac{1}{\sin(\theta)} \) and \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \), calculate \( \sin(\theta) = \frac{\sqrt{3}}{2\sqrt{3}} \).

Finally, express \( \csc(\theta) \) as the reciprocal of \( \sin(\theta) \), which is \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Simplify this expression 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.

Inverse Trigonometric Functions

Inverse trigonometric functions, like tan⁻¹(x), return the angle whose trigonometric ratio equals x. For example, tan⁻¹(√3/3) gives the angle whose tangent is √3/3, which helps in finding the angle measure needed to evaluate other trigonometric functions.

Right Triangle Trigonometry

Right triangle trigonometry relates the sides of a right triangle to its angles using ratios like sine, cosine, and tangent. Knowing the tangent ratio allows you to determine the opposite and adjacent sides, which can then be used to find other ratios such as cosecant.

Reciprocal Trigonometric Functions

Cosecant (csc) is the reciprocal of sine, defined as csc(θ) = 1/sin(θ). Once the angle θ is found, calculating csc(θ) involves finding the sine of θ and then taking its reciprocal to get the exact value.