cos^5(x)= 1/2

{ "query": { "display": "$$\\cos^{5}\\left(x\\right)=\\frac{1}{2}$$", "symbolab_question": "EQUATION#\\cos^{5}(x)=\\frac{1}{2}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=0.51447…+2πn,x=2π-0.51447…+2πn", "degrees": "x=29.47731…^{\\circ }+360^{\\circ }n,x=330.52268…^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos^{5}\\left(x\\right)=\\frac{1}{2}{\\quad:\\quad}x=0.51447…+2πn,\\:x=2π-0.51447…+2πn$$", "input": "\\cos^{5}\\left(x\\right)=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "For $$x^n=f\\left(a\\right)$$, n is odd, the solution is $$x=\\sqrt[n]{f\\left(a\\right)}$$" }, { "type": "step", "result": "\\cos\\left(x\\right)=\\sqrt[5]{\\frac{1}{2}}" }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cos\\left(x\\right)=\\sqrt[5]{\\frac{1}{2}}", "result": "x=\\arccos\\left(\\sqrt[5]{\\frac{1}{2}}\\right)+2πn,\\:x=2π-\\arccos\\left(\\sqrt[5]{\\frac{1}{2}}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cos\\left(x\\right)=\\sqrt[5]{\\frac{1}{2}}$$", "secondary": [ "$$\\cos\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arccos\\left(a\\right)+2πn,\\:\\quad\\:x=2π-\\arccos\\left(a\\right)+2πn$$" ], "result": "x=\\arccos\\left(\\sqrt[5]{\\frac{1}{2}}\\right)+2πn,\\:x=2π-\\arccos\\left(\\sqrt[5]{\\frac{1}{2}}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "step", "primary": "Show solutions in decimal form", "result": "x=0.51447…+2πn,\\:x=2π-0.51447…+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\cos^{5}(x)-\\frac{1}{2}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }