{ "query": { "display": "$$\\sin\\left(a\\right)=\\frac{3}{5},\\:\\cos\\left(a\\right)$$", "symbolab_question": "EQUATION#\\sin(a)=\\frac{3}{5},\\cos(a)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "\\mathrm{No\\:Solution\\:for}\\:a\\in\\mathbb{R}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(a\\right)=\\frac{3}{5},\\:\\cos\\left(a\\right){\\quad:\\quad}$$No Solution for $$a\\in\\mathbb{R}$$", "input": "\\sin\\left(a\\right)=\\frac{3}{5},\\:\\cos\\left(a\\right)", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(a\\right)=\\frac{3}{5}", "result": "a=\\arcsin\\left(\\frac{3}{5}\\right)+2πn,\\:a=π-\\arcsin\\left(\\frac{3}{5}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(a\\right)=\\frac{3}{5}$$", "secondary": [ "$$\\sin\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arcsin\\left(a\\right)+2πn,\\:\\quad\\:x=π-\\arcsin\\left(a\\right)+2πn$$" ], "result": "a=\\arcsin\\left(\\frac{3}{5}\\right)+2πn,\\:a=π-\\arcsin\\left(\\frac{3}{5}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "step", "primary": "Solutions for the range $$\\cos\\left(a\\right)$$", "result": "\\mathrm{No\\:Solution\\:for}\\:a\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "a", "plotRequest": "\\sin(a)-\\frac{3}{5},\\cos(a)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }