Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. tan ( 5𝝅/3 ﹣ 𝝅/4)

검증된 단계별 안내

Identify the given expression: \(\tan\left( \frac{5\pi}{3} - \frac{\pi}{4} \right)\).

Recall the tangent difference identity: \(\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\).

Set \(A = \frac{5\pi}{3}\) and \(B = \frac{\pi}{4}\), then find \(\tan A\) and \(\tan B\) separately.

Calculate \(\tan \frac{5\pi}{3}\) and \(\tan \frac{\pi}{4}\) using known values or reference angles.

Substitute these values into the tangent difference formula and simplify the resulting expression step-by-step.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

주요 개념

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

Sum and Difference Identities for Tangent

These identities express the tangent of a sum or difference of two angles in terms of the tangents of the individual angles. Specifically, tan(A - B) = (tan A - tan B) / (1 + tan A * tan B). This formula is essential for breaking down complex tangent expressions into simpler parts.

Exact Values of Trigonometric Functions at Special Angles

Certain angles like π/3, π/4, and π/5 have known exact trigonometric values involving square roots and rational numbers. Knowing these values allows for precise calculation without approximations, which is crucial when solving problems requiring exact answers.

Simplification of Trigonometric Expressions

After applying identities, simplifying the resulting expressions by combining like terms, rationalizing denominators, or reducing fractions is necessary. This step ensures the final answer is in its simplest exact form, which is often required in trigonometry exercises.