sin^2((17pi)/2)+cos^2((17pi)/2)

{ "query": { "display": "$$\\sin^{2}\\left(\\frac{17π}{2}\\right)+\\cos^{2}\\left(\\frac{17π}{2}\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\sin^{2}(\\frac{17π}{2})+\\cos^{2}(\\frac{17π}{2})" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "1", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin^{2}\\left(\\frac{17π}{2}\\right)+\\cos^{2}\\left(\\frac{17π}{2}\\right)=1$$", "input": "\\sin^{2}\\left(\\frac{17π}{2}\\right)+\\cos^{2}\\left(\\frac{17π}{2}\\right)", "steps": [ { "type": "interim", "title": "$$\\sin\\left(\\frac{17π}{2}\\right)=\\sin\\left(\\frac{π}{2}\\right)$$", "input": "\\sin\\left(\\frac{17π}{2}\\right)", "result": "=\\sin^{2}\\left(\\frac{π}{2}\\right)+\\cos^{2}\\left(\\frac{17π}{2}\\right)", "steps": [ { "type": "step", "primary": "Rewrite $$\\frac{17π}{2}$$ as $$2π\\cdot\\:4+\\frac{π}{2}$$", "result": "=\\sin\\left(2π4+\\frac{π}{2}\\right)" }, { "type": "step", "primary": "Apply the periodicity of $$\\sin$$: $$\\sin\\left(x+2π\\cdot\\:k\\right)=\\sin\\left(x\\right)$$", "secondary": [ "$$\\sin\\left(2π\\cdot\\:4+\\frac{π}{2}\\right)=\\sin\\left(\\frac{π}{2}\\right)$$" ], "result": "=\\sin\\left(\\frac{π}{2}\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(\\frac{17π}{2}\\right)=\\cos\\left(\\frac{π}{2}\\right)$$", "input": "\\cos\\left(\\frac{17π}{2}\\right)", "result": "=\\sin^{2}\\left(\\frac{π}{2}\\right)+\\cos^{2}\\left(\\frac{π}{2}\\right)", "steps": [ { "type": "step", "primary": "Rewrite $$\\frac{17π}{2}$$ as $$2π\\cdot\\:4+\\frac{π}{2}$$", "result": "=\\cos\\left(2π4+\\frac{π}{2}\\right)" }, { "type": "step", "primary": "Apply the periodicity of $$\\cos$$: $$\\cos\\left(x+2π\\cdot\\:k\\right)=\\cos\\left(x\\right)$$", "secondary": [ "$$\\cos\\left(2π\\cdot\\:4+\\frac{π}{2}\\right)=\\cos\\left(\\frac{π}{2}\\right)$$" ], "result": "=\\cos\\left(\\frac{π}{2}\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(\\frac{π}{2}\\right)=1$$", "input": "\\sin\\left(\\frac{π}{2}\\right)", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=1" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(\\frac{π}{2}\\right)=0$$", "input": "\\cos\\left(\\frac{π}{2}\\right)", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=0" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=1^{2}+0^{2}" }, { "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 } }