cot((2pi)/3)-sin((3pi)/2)

{ "query": { "display": "$$\\cot\\left(\\frac{2π}{3}\\right)-\\sin\\left(\\frac{3π}{2}\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\cot(\\frac{2π}{3})-\\sin(\\frac{3π}{2})" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "-\\frac{\\sqrt{3}}{3}+1", "decimal": "0.42264…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cot\\left(\\frac{2π}{3}\\right)-\\sin\\left(\\frac{3π}{2}\\right)=-\\frac{\\sqrt{3}}{3}+1$$", "input": "\\cot\\left(\\frac{2π}{3}\\right)-\\sin\\left(\\frac{3π}{2}\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\sin\\left(\\frac{3π}{2}\\right)=2\\sin\\left(\\frac{3π}{4}\\right)\\cos\\left(\\frac{3π}{4}\\right)$$", "input": "\\sin\\left(\\frac{3π}{2}\\right)", "result": "=\\cot\\left(\\frac{2π}{3}\\right)-2\\sin\\left(\\frac{3π}{4}\\right)\\cos\\left(\\frac{3π}{4}\\right)", "steps": [ { "type": "step", "primary": "Write $$\\sin\\left(\\frac{3π}{2}\\right)\\:$$as $$\\sin\\left(2\\cdot\\:\\frac{3π}{4}\\right)$$", "result": "=\\sin\\left(2\\cdot\\:\\frac{3π}{4}\\right)" }, { "type": "step", "primary": "Use the Double Angle identity: $$\\sin\\left(2x\\right)=2\\sin\\left(x\\right)\\cos\\left(x\\right)$$", "result": "=2\\sin\\left(\\frac{3π}{4}\\right)\\cos\\left(\\frac{3π}{4}\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cot\\left(\\frac{2π}{3}\\right)=-\\frac{\\sqrt{3}}{3}$$", "input": "\\cot\\left(\\frac{2π}{3}\\right)", "steps": [ { "type": "step", "primary": "$$\\cot\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\cot(x)\\\\\\hline 0&\\mp\\infty\\\\\\hline \\frac{π}{6}&\\sqrt{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\sqrt{3}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=-\\frac{\\sqrt{3}}{3}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(\\frac{3π}{4}\\right)=\\frac{\\sqrt{2}}{2}$$", "input": "\\sin\\left(\\frac{3π}{4}\\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": "=\\frac{\\sqrt{2}}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(\\frac{3π}{4}\\right)=-\\frac{\\sqrt{2}}{2}$$", "input": "\\cos\\left(\\frac{3π}{4}\\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": "=-\\frac{\\sqrt{2}}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=-\\frac{\\sqrt{3}}{3}-2\\cdot\\:\\frac{\\sqrt{2}}{2}\\left(-\\frac{\\sqrt{2}}{2}\\right)" }, { "type": "interim", "title": "Simplify $$-\\frac{\\sqrt{3}}{3}-2\\cdot\\:\\frac{\\sqrt{2}}{2}\\left(-\\frac{\\sqrt{2}}{2}\\right):{\\quad}-\\frac{\\sqrt{3}}{3}+1$$", "input": "-\\frac{\\sqrt{3}}{3}-2\\cdot\\:\\frac{\\sqrt{2}}{2}\\left(-\\frac{\\sqrt{2}}{2}\\right)", "result": "=-\\frac{\\sqrt{3}}{3}+1", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=-\\frac{\\sqrt{3}}{3}+2\\cdot\\:\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{\\sqrt{2}}{2}" }, { "type": "interim", "title": "$$2\\cdot\\:\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{\\sqrt{2}}{2}=1$$", "input": "2\\cdot\\:\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{\\sqrt{2}}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}\\cdot\\frac{d}{e}=\\frac{a\\:\\cdot\\:b\\:\\cdot\\:d}{c\\:\\cdot\\:e}$$", "result": "=\\frac{\\sqrt{2}\\sqrt{2}\\cdot\\:2}{2\\cdot\\:2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{\\sqrt{2}\\sqrt{2}}{2}" }, { "type": "interim", "title": "$$\\sqrt{2}\\sqrt{2}=2$$", "input": "\\sqrt{2}\\sqrt{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{2}\\sqrt{2}=2$$" ], "result": "=2", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OXRSLY4uXiMbvMfICP5T88aFSKFknkW7qg0U6E21MorMwViaLUXkeD+JukROhWdjBJz+17+I6kHbIEeqwhy4mdC5tTYOsrqRy+4iBsRgZTQLmgWGjI523GgzSlA2k6Cv" } }, { "type": "step", "result": "=\\frac{2}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvihqxDwJ92JV1QKn+tjCIbSEJr9fcKtGOvM3Tu2Nnvco3yhf7NjUFA9zUXJlwDFbstgOfOVs9mPIqDLV5QIWwt3mXUrWvRppVP8TTNlqQgP2MtkSd66Y+6btTv2/ygc3ud3pO5mI2mZYNCL62n3DPvLNE1DBYBMU/d/MOZ1Ocewpobulg1IX7L1yvFBxGjhZxJ7CI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=-\\frac{\\sqrt{3}}{3}+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7rpPPo+Nmcur7TLc75qZWRdC682G3P2hqfW/Jb3GEULMussnnnisJndCAm4z+zno87ww4i5CqvzmdiRlWd3LwB+s/IHIBptIRx5iJkqAvACTdd47a0hQ8flDbGsI5To1dsabmmR9ZBQ+yLvQJaWqH0L4RhgczeMv19ftoUw2cH9RkS3dlcCKpQTQcheuut7MkdkqXaZXgCXO2IClMAWKDPSAB+riJVcmOQWxDhaDbqK/QuvNhtz9oan1vyW9xhFCzLrLJ554rCZ3QgJuM/s56PO8MOIuQqr85nYkZVndy8AfiqBmhol4UCcrZiJeso7O+" } } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }