sin(990)

{ "query": { "display": "$$\\sin\\left(990^{\\circ\\:}\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\sin(990^{\\circ })" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "-1", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(990^{\\circ\\:}\\right)=-1$$", "input": "\\sin\\left(990^{\\circ\\:}\\right)", "steps": [ { "type": "interim", "title": "$$\\sin\\left(990^{\\circ\\:}\\right)=\\sin\\left(270^{\\circ\\:}\\right)$$", "input": "\\sin\\left(990^{\\circ\\:}\\right)", "result": "=\\sin\\left(270^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "Rewrite $$990^{\\circ\\:}$$ as $$360^{\\circ\\:}\\cdot\\:2+270^{\\circ\\:}$$", "result": "=\\sin\\left(360^{\\circ\\:}2+270^{\\circ\\:}\\right)" }, { "type": "step", "primary": "Apply the periodicity of $$\\sin$$: $$\\sin\\left(x+360^{\\circ\\:}\\cdot\\:k\\right)=\\sin\\left(x\\right)$$", "secondary": [ "$$\\sin\\left(360^{\\circ\\:}\\cdot\\:2+270^{\\circ\\:}\\right)=\\sin\\left(270^{\\circ\\:}\\right)$$" ], "result": "=\\sin\\left(270^{\\circ\\:}\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\sin\\left(180^{\\circ\\:}\\right)\\cos\\left(90^{\\circ\\:}\\right)+\\cos\\left(180^{\\circ\\:}\\right)\\sin\\left(90^{\\circ\\:}\\right)$$", "input": "\\sin\\left(270^{\\circ\\:}\\right)", "result": "=\\sin\\left(180^{\\circ\\:}\\right)\\cos\\left(90^{\\circ\\:}\\right)+\\cos\\left(180^{\\circ\\:}\\right)\\sin\\left(90^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "Write $$\\sin\\left(270^{\\circ\\:}\\right)\\:$$as $$\\sin\\left(180^{\\circ\\:}+90^{\\circ\\:}\\right)$$", "result": "=\\sin\\left(180^{\\circ\\:}+90^{\\circ\\:}\\right)" }, { "type": "step", "primary": "Use the Angle Sum identity: $$\\sin\\left(s+t\\right)=\\sin\\left(s\\right)\\cos\\left(t\\right)+\\cos\\left(s\\right)\\sin\\left(t\\right)$$", "result": "=\\sin\\left(180^{\\circ\\:}\\right)\\cos\\left(90^{\\circ\\:}\\right)+\\cos\\left(180^{\\circ\\:}\\right)\\sin\\left(90^{\\circ\\:}\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(180^{\\circ\\:}\\right)=0$$", "input": "\\sin\\left(180^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$360^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&180^{\\circ }&0\\\\\\hline 30^{\\circ }&\\frac{1}{2}&210^{\\circ }&-\\frac{1}{2}\\\\\\hline 45^{\\circ }&\\frac{\\sqrt{2}}{2}&225^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 60^{\\circ }&\\frac{\\sqrt{3}}{2}&240^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 90^{\\circ }&1&270^{\\circ }&-1\\\\\\hline 120^{\\circ }&\\frac{\\sqrt{3}}{2}&300^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 135^{\\circ }&\\frac{\\sqrt{2}}{2}&315^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 150^{\\circ }&\\frac{1}{2}&330^{\\circ }&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=0" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(90^{\\circ\\:}\\right)=0$$", "input": "\\cos\\left(90^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$360^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&180^{\\circ }&-1\\\\\\hline 30^{\\circ }&\\frac{\\sqrt{3}}{2}&210^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 45^{\\circ }&\\frac{\\sqrt{2}}{2}&225^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 60^{\\circ }&\\frac{1}{2}&240^{\\circ }&-\\frac{1}{2}\\\\\\hline 90^{\\circ }&0&270^{\\circ }&0\\\\\\hline 120^{\\circ }&-\\frac{1}{2}&300^{\\circ }&\\frac{1}{2}\\\\\\hline 135^{\\circ }&-\\frac{\\sqrt{2}}{2}&315^{\\circ }&\\frac{\\sqrt{2}}{2}\\\\\\hline 150^{\\circ }&-\\frac{\\sqrt{3}}{2}&330^{\\circ }&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=0" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(180^{\\circ\\:}\\right)=\\left(-1\\right)$$", "input": "\\cos\\left(180^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$360^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&180^{\\circ }&-1\\\\\\hline 30^{\\circ }&\\frac{\\sqrt{3}}{2}&210^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 45^{\\circ }&\\frac{\\sqrt{2}}{2}&225^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 60^{\\circ }&\\frac{1}{2}&240^{\\circ }&-\\frac{1}{2}\\\\\\hline 90^{\\circ }&0&270^{\\circ }&0\\\\\\hline 120^{\\circ }&-\\frac{1}{2}&300^{\\circ }&\\frac{1}{2}\\\\\\hline 135^{\\circ }&-\\frac{\\sqrt{2}}{2}&315^{\\circ }&\\frac{\\sqrt{2}}{2}\\\\\\hline 150^{\\circ }&-\\frac{\\sqrt{3}}{2}&330^{\\circ }&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=\\left(-1\\right)" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(90^{\\circ\\:}\\right)=1$$", "input": "\\sin\\left(90^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$360^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&180^{\\circ }&0\\\\\\hline 30^{\\circ }&\\frac{1}{2}&210^{\\circ }&-\\frac{1}{2}\\\\\\hline 45^{\\circ }&\\frac{\\sqrt{2}}{2}&225^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 60^{\\circ }&\\frac{\\sqrt{3}}{2}&240^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 90^{\\circ }&1&270^{\\circ }&-1\\\\\\hline 120^{\\circ }&\\frac{\\sqrt{3}}{2}&300^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 135^{\\circ }&\\frac{\\sqrt{2}}{2}&315^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 150^{\\circ }&\\frac{1}{2}&330^{\\circ }&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=1" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=0\\cdot\\:0+\\left(-1\\right)\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=-1" } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }