solve equation, give a general formula for all solutions

(2 cos x - sqrt{2} = 0)

Question

solve equation, give a general formula for all solutions

(2 cos x - sqrt{2} = 0)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Trigonometric Equations" ], "new_concepts": [], "current_concepts": [ "Trigonometric Equations" ] } ## Isolate the cosine term $$ \begin{aligned} 2\cos x - \sqrt{2} &= 0 \ 2\cos x &= \sqrt{2} \ \cos x &= \frac{\sqrt{2}}{2} \end{aligned} $$ ## Find the principal solutions $$ \begin{aligned} x &= \frac{\pi}{4} \ x &= -\frac{\pi}{4} \quad (\text{or } \frac{7\pi}{4}) \end{aligned} $$ ## Write the general formula $$ x = \pm\frac{\pi}{4} + 2k\pi, \quad k \in \mathbb{Z} $$ $$x = \pm\frac{\pi}{4} + 2k\pi, \quad k \in \mathbb{Z}$$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Trigonometric Equations" ] }

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Question

Explanation:

Isolate the cosine term

Find the principal solutions

Write the general formula

Answer: