cos(8t)-5sin(8t)=0

{ "query": { "display": "$$\\cos\\left(8t\\right)-5\\sin\\left(8t\\right)=0$$", "symbolab_question": "EQUATION#\\cos(8t)-5\\sin(8t)=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "t=\\frac{0.19739…}{8}+\\frac{πn}{8}", "degrees": "t=1.41374…^{\\circ }+22.5^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(8t\\right)-5\\sin\\left(8t\\right)=0{\\quad:\\quad}t=\\frac{0.19739…}{8}+\\frac{πn}{8}$$", "input": "\\cos\\left(8t\\right)-5\\sin\\left(8t\\right)=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\cos\\left(8t\\right)-5\\sin\\left(8t\\right)=0", "result": "1-5\\tan\\left(8t\\right)=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$\\cos\\left(8t\\right),\\:\\cos\\left(8t\\right)\\neq0$$", "result": "\\frac{\\cos\\left(8t\\right)-5\\sin\\left(8t\\right)}{\\cos\\left(8t\\right)}=\\frac{0}{\\cos\\left(8t\\right)}" }, { "type": "step", "primary": "Simplify", "result": "1-\\frac{5\\sin\\left(8t\\right)}{\\cos\\left(8t\\right)}=0" }, { "type": "step", "primary": "Use the basic trigonometric identity: $$\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}=\\tan\\left(x\\right)$$", "result": "1-5\\tan\\left(8t\\right)=0" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lRPp6QndqgNT+L6IZutAbB/8jntDa3QhUepw1clbpHMaM37KF/jK0q16Pg+4JBw35yFOVAzu8w4itJwsMopep5nX/LTRLwEhE6Awmk/6zhWTs+QHNBcNmX4YsFPSKTfdeT6jUBOkw/FGPCyV4AShke9sGZu5A1MXROmEpnxG69oGqP+/Ugc9TvNWR3IfFZuE9mth+xoYA4yaoGna981gpoZ+y+YEwVMxeIPlP7C0eko=" } }, { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "1-5\\tan\\left(8t\\right)=0", "result": "-5\\tan\\left(8t\\right)=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "1-5\\tan\\left(8t\\right)-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "-5\\tan\\left(8t\\right)=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-5$$", "input": "-5\\tan\\left(8t\\right)=-1", "result": "\\tan\\left(8t\\right)=\\frac{1}{5}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-5$$", "result": "\\frac{-5\\tan\\left(8t\\right)}{-5}=\\frac{-1}{-5}" }, { "type": "step", "primary": "Simplify", "result": "\\tan\\left(8t\\right)=\\frac{1}{5}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\tan\\left(8t\\right)=\\frac{1}{5}", "result": "8t=\\arctan\\left(\\frac{1}{5}\\right)+πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\tan\\left(8t\\right)=\\frac{1}{5}$$", "secondary": [ "$$\\tan\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arctan\\left(a\\right)+πn$$" ], "result": "8t=\\arctan\\left(\\frac{1}{5}\\right)+πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "interim", "title": "Solve $$8t=\\arctan\\left(\\frac{1}{5}\\right)+πn:{\\quad}t=\\frac{\\arctan\\left(\\frac{1}{5}\\right)}{8}+\\frac{πn}{8}$$", "input": "8t=\\arctan\\left(\\frac{1}{5}\\right)+πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$8$$", "input": "8t=\\arctan\\left(\\frac{1}{5}\\right)+πn", "result": "t=\\frac{\\arctan\\left(\\frac{1}{5}\\right)}{8}+\\frac{πn}{8}", "steps": [ { "type": "step", "primary": "Divide both sides by $$8$$", "result": "\\frac{8t}{8}=\\frac{\\arctan\\left(\\frac{1}{5}\\right)}{8}+\\frac{πn}{8}" }, { "type": "step", "primary": "Simplify", "result": "t=\\frac{\\arctan\\left(\\frac{1}{5}\\right)}{8}+\\frac{πn}{8}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "t=\\frac{\\arctan\\left(\\frac{1}{5}\\right)}{8}+\\frac{πn}{8}" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "t=\\frac{0.19739…}{8}+\\frac{πn}{8}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "t", "plotRequest": "\\cos(8t)-5\\sin(8t)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }