solvefor a,P=cot^2(a)

{ "query": { "display": "solve for $$a,\\:P=\\cot^{2}\\left(a\\right)$$", "symbolab_question": "SOLVE_FOR#solvefor a,P=\\cot^{2}(a)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "a=\\arccot(\\sqrt{P})+πn,a=\\arccot(-\\sqrt{P})+πn", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$solve\\:for\\:a,\\:P=\\cot^{2}\\left(a\\right){\\quad:\\quad}a=\\arccot\\left(\\sqrt{P}\\right)+πn,\\:a=\\arccot\\left(-\\sqrt{P}\\right)+πn$$", "input": "P=\\cot^{2}\\left(a\\right)", "steps": [ { "type": "step", "primary": "Switch sides", "result": "\\cot^{2}\\left(a\\right)=P" }, { "type": "interim", "title": "Solve by substitution", "input": "\\cot^{2}\\left(a\\right)=P", "result": "\\cot\\left(a\\right)=\\sqrt{P},\\:\\cot\\left(a\\right)=-\\sqrt{P}", "steps": [ { "type": "step", "primary": "Let: $$\\cot\\left(a\\right)=u$$", "result": "u^{2}=P" }, { "type": "interim", "title": "$$u^{2}=P{\\quad:\\quad}u=\\sqrt{P},\\:u=-\\sqrt{P}$$", "input": "u^{2}=P", "steps": [ { "type": "step", "primary": "For $$x^{2}=f\\left(a\\right)$$ the solutions are $$x=\\sqrt{f\\left(a\\right)},\\:\\:-\\sqrt{f\\left(a\\right)}$$" }, { "type": "step", "result": "u=\\sqrt{P},\\:u=-\\sqrt{P}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cot\\left(a\\right)$$", "result": "\\cot\\left(a\\right)=\\sqrt{P},\\:\\cot\\left(a\\right)=-\\sqrt{P}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cot\\left(a\\right)=\\sqrt{P}{\\quad:\\quad}a=\\arccot\\left(\\sqrt{P}\\right)+πn$$", "input": "\\cot\\left(a\\right)=\\sqrt{P}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cot\\left(a\\right)=\\sqrt{P}", "result": "a=\\arccot\\left(\\sqrt{P}\\right)+πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cot\\left(a\\right)=\\sqrt{P}$$", "secondary": [ "$$\\cot\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arccot\\left(a\\right)+πn$$" ], "result": "a=\\arccot\\left(\\sqrt{P}\\right)+πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cot\\left(a\\right)=-\\sqrt{P}{\\quad:\\quad}a=\\arccot\\left(-\\sqrt{P}\\right)+πn$$", "input": "\\cot\\left(a\\right)=-\\sqrt{P}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cot\\left(a\\right)=-\\sqrt{P}", "result": "a=\\arccot\\left(-\\sqrt{P}\\right)+πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cot\\left(a\\right)=-\\sqrt{P}$$", "secondary": [ "$$\\cot\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arccot\\left(a\\right)+πn$$" ], "result": "a=\\arccot\\left(-\\sqrt{P}\\right)+πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "a=\\arccot\\left(\\sqrt{P}\\right)+πn,\\:a=\\arccot\\left(-\\sqrt{P}\\right)+πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "a", "plotRequest": "P-\\cot^{2}(a)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }