Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. See Example 6. sec θ = -√2

검증된 단계별 안내

Recall the definition of secant: \(\sec \theta = \frac{1}{\cos \theta}\). So, the equation \(\sec \theta = -\sqrt{2}\) can be rewritten as \(\frac{1}{\cos \theta} = -\sqrt{2}\).

Solve for \(\cos \theta\) by taking the reciprocal of both sides: \(\cos \theta = -\frac{1}{\sqrt{2}}\).

Recognize that \(\cos \theta = -\frac{1}{\sqrt{2}}\) is equivalent to \(\cos \theta = -\frac{\sqrt{2}}{2}\) after rationalizing the denominator.

Determine the reference angle where \(\cos \theta = \frac{\sqrt{2}}{2}\). This reference angle is \(45^\circ\) because \(\cos 45^\circ = \frac{\sqrt{2}}{2}\).

Since \(\cos \theta\) is negative, find all angles in the interval \([0^\circ, 360^\circ)\) where cosine is negative. Cosine is negative in the second and third quadrants, so the solutions are \(\theta = 180^\circ - 45^\circ\) and \(\theta = 180^\circ + 45^\circ\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

Definition of Secant Function

The secant function, sec θ, is the reciprocal of the cosine function, defined as sec θ = 1/cos θ. Understanding this relationship allows us to convert secant equations into cosine equations, which are often easier to solve.

Solving Trigonometric Equations in a Given Interval

When solving trigonometric equations like sec θ = -√2 over [0°, 360°), it is essential to find all angles θ within the interval that satisfy the equation. This involves considering the periodicity and sign of the trigonometric functions in different quadrants.

Sign of Trigonometric Functions in Quadrants

The sign of cosine (and thus secant) varies by quadrant: cosine is positive in the first and fourth quadrants and negative in the second and third. Since sec θ = 1/cos θ, secant shares the same sign pattern, which helps identify the correct quadrants for solutions.