In Exercises 49–59, find the exact value of each expression. Do not use a calculator. sec 7𝜋 / 4

검증된 단계별 안내

Recall the definition of secant: \(\sec \theta = \frac{1}{\cos \theta}\). So, to find \(\sec \frac{7\pi}{4}\), we first need to find \(\cos \frac{7\pi}{4}\).

Identify the angle \(\frac{7\pi}{4}\) on the unit circle. Since \(\frac{7\pi}{4}\) is between \(2\pi\) and \(\frac{3\pi}{2}\), find its reference angle by subtracting from \(2\pi\): \(2\pi - \frac{7\pi}{4} = \frac{\pi}{4}\).

Determine the cosine value of the reference angle \(\frac{\pi}{4}\). We know \(\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\).

Determine the sign of cosine in the quadrant where \(\frac{7\pi}{4}\) lies. Since \(\frac{7\pi}{4}\) is in the fourth quadrant, cosine is positive there, so \(\cos \frac{7\pi}{4} = \frac{\sqrt{2}}{2}\).

Finally, use the secant definition to write \(\sec \frac{7\pi}{4} = \frac{1}{\cos \frac{7\pi}{4}} = \frac{1}{\frac{\sqrt{2}}{2}}\). Simplify this expression to get 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.

Understanding the Secant Function

The secant function, sec(θ), is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). To find sec(7π/4), you first find cos(7π/4) and then take its reciprocal. This relationship is fundamental for evaluating secant values without a calculator.

Reference Angles and Unit Circle Values

Angles like 7π/4 can be related to reference angles within the first quadrant to determine their trigonometric values. Since 7π/4 is in the fourth quadrant, its reference angle is π/4. Knowing the cosine value at π/4 and the sign of cosine in the fourth quadrant helps find cos(7π/4) exactly.

Sign of Trigonometric Functions in Quadrants

The sign of cosine and secant depends on the quadrant of the angle. In the fourth quadrant (where 7π/4 lies), cosine is positive, so secant will also be positive. Understanding the ASTC (All Students Take Calculus) rule helps determine the correct sign of the trigonometric value.