sin(30)+cos(45)

{ "query": { "display": "$$\\sin\\left(30^{\\circ\\:}\\right)+\\cos\\left(45^{\\circ\\:}\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\sin(30^{\\circ })+\\cos(45^{\\circ })" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "\\frac{1+\\sqrt{2}}{2}", "decimal": "1.20710…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(30^{\\circ\\:}\\right)+\\cos\\left(45^{\\circ\\:}\\right)=\\frac{1+\\sqrt{2}}{2}$$", "input": "\\sin\\left(30^{\\circ\\:}\\right)+\\cos\\left(45^{\\circ\\:}\\right)", "steps": [ { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(30^{\\circ\\:}\\right)=\\frac{1}{2}$$", "input": "\\sin\\left(30^{\\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": "=\\frac{1}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(45^{\\circ\\:}\\right)=\\frac{\\sqrt{2}}{2}$$", "input": "\\cos\\left(45^{\\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": "=\\frac{\\sqrt{2}}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=\\frac{1}{2}+\\frac{\\sqrt{2}}{2}" }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}+\\frac{\\sqrt{2}}{2}:{\\quad}\\frac{1+\\sqrt{2}}{2}$$", "input": "\\frac{1}{2}+\\frac{\\sqrt{2}}{2}", "result": "=\\frac{1+\\sqrt{2}}{2}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{1+\\sqrt{2}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/IqmB+KRU+T1KryXQuitcXpO5mI2mZYNCL62n3DPvLMnVQZ9jTQhWaoOGaFYelMKo5FYteSPKwXny4uCMrdsK+FOuoiK/FEPJUJdZ04AhzsaZ96EGr8gUJUjHsZKSX6rHjb2+5NLFZrsH9fcPWg/TcBZCZf4UBlDnt5DwfBsIkKpiJ2dUZBdrTH+MQdA7imTrxRYDHW4QXgWVqktVCHQCw==" } } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }