2arcsec(2)+arcsin(1/2)

{ "query": { "display": "$$2\\arcsec\\left(2\\right)+\\arcsin\\left(\\frac{1}{2}\\right)$$", "symbolab_question": "TRIG_EVALUATE#2\\arcsec(2)+\\arcsin(\\frac{1}{2})" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "\\frac{5π}{6}", "decimal": "150", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2\\arcsec\\left(2\\right)+\\arcsin\\left(\\frac{1}{2}\\right)=\\frac{5π}{6}$$", "input": "2\\arcsec\\left(2\\right)+\\arcsin\\left(\\frac{1}{2}\\right)", "steps": [ { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\arcsec\\left(2\\right)=\\frac{π}{3}$$", "input": "\\arcsec\\left(2\\right)", "steps": [ { "type": "step", "primary": "$$\\begin{array}{|c|c|c|}\\hline x&\\arcsec(x)&\\arcsec(x)\\\\\\hline 1&0&0^{\\circ}\\\\\\hline \\frac{2\\sqrt{3}}{3}&\\frac{\\pi}{6}&30^{\\circ}\\\\\\hline \\sqrt{2}&\\frac{\\pi}{4}&45^{\\circ}\\\\\\hline 2&\\frac{\\pi}{3}&60^{\\circ}\\\\\\hline -2&\\frac{2\\pi}{3}&120^{\\circ}\\\\\\hline -\\sqrt{2}&\\frac{3\\pi}{4}&135^{\\circ}\\\\\\hline -\\frac{2\\sqrt{3}}{3}&\\frac{5\\pi}{6}&150^{\\circ}\\\\\\hline -1&\\pi&180^{\\circ}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=\\frac{π}{3}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\arcsin\\left(\\frac{1}{2}\\right)=\\frac{π}{6}$$", "input": "\\arcsin\\left(\\frac{1}{2}\\right)", "steps": [ { "type": "step", "primary": "$$\\begin{array}{|c|c|c|}\\hline x&\\arcsin(x)&\\arcsin(x)\\\\\\hline 0&0&0^{\\circ}\\\\\\hline \\frac{1}{2}&\\frac{\\pi}{6}&30^{\\circ}\\\\\\hline \\frac{\\sqrt{2}}{2}&\\frac{\\pi}{4}&45^{\\circ}\\\\\\hline \\frac{\\sqrt{3}}{2}&\\frac{\\pi}{3}&60^{\\circ}\\\\\\hline 1&\\frac{\\pi}{2}&90^{\\circ}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=\\frac{π}{6}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=2\\cdot\\:\\frac{π}{3}+\\frac{π}{6}" }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{π}{3}+\\frac{π}{6}:{\\quad}\\frac{5π}{6}$$", "input": "2\\cdot\\:\\frac{π}{3}+\\frac{π}{6}", "result": "=\\frac{5π}{6}", "steps": [ { "type": "interim", "title": "Multiply $$2\\cdot\\:\\frac{π}{3}\\::{\\quad}\\frac{2π}{3}$$", "input": "2\\cdot\\:\\frac{π}{3}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{π2}{3}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "result": "=\\frac{2π}{3}+\\frac{π}{6}" }, { "type": "interim", "title": "Least Common Multiplier of $$3,\\:6:{\\quad}6$$", "input": "3,\\:6", "steps": [ { "type": "definition", "title": "Least Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b$$ is the smallest positive number that is divisible by both $$a$$ and $$b$$" }, { "type": "interim", "title": "Prime factorization of $$3:{\\quad}3$$", "input": "3", "steps": [ { "type": "step", "primary": "$$3$$ is a prime number, therefore no factorization is possible", "result": "=3" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRlqnfsqoQ6VBiS8EyG3E6Oc/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp39nbJbLYrlgLb4BA6ndvX8" } }, { "type": "interim", "title": "Prime factorization of $$6:{\\quad}2\\cdot\\:3$$", "input": "6", "steps": [ { "type": "step", "primary": "$$6\\:$$divides by $$2\\quad\\:6=3\\cdot\\:2$$", "result": "=2\\cdot\\:3" }, { "type": "step", "primary": "$$2,\\:3$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:3" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRuUHkFwKrCGUG/pR2kioRow/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp1AjXz67i9oO9i25G22wINi" } }, { "type": "step", "primary": "Multiply each factor the greatest number of times it occurs in either $$3$$ or $$6$$", "result": "=3\\cdot\\:2" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=6" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top 1Eq" } }, { "type": "interim", "title": "Adjust Fractions based on the LCM", "steps": [ { "type": "step", "primary": "Multiply each numerator by the same amount needed to multiply its<br/>corresponding denominator to turn it into the LCM $$6$$" }, { "type": "step", "primary": "For $$\\frac{π2}{3}:\\:$$multiply the denominator and numerator by $$2$$", "result": "\\frac{π2}{3}=\\frac{π2\\cdot\\:2}{3\\cdot\\:2}=\\frac{4π}{6}" } ], "meta": { "interimType": "LCD Adjust Fractions 1Eq" } }, { "type": "step", "result": "=\\frac{4π}{6}+\\frac{π}{6}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{4π+π}{6}" }, { "type": "step", "primary": "Add similar elements: $$4π+π=5π$$", "result": "=\\frac{5π}{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvihXNyjOm7+riaxLrdK/l4JRTIvmwy5GMzS71gO2z89sScJChiVhDxT5N/LHSTLMjyCEhfs2SCN7/OIlOoRO3Fl0/y9DKGIPglJ+qMi9xDu2KaRI7GCp0HQz+zDw23axddLxB6YWVlteWqXIBjnoremFm7RiMpMPpe+SF9pfwjIMhPJ8UXOwXx1x/ECg7VWlZDQ==" } } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }