5/(tan(30))

{ "query": { "display": "$$\\frac{5}{\\tan\\left(30^{\\circ\\:}\\right)}$$", "symbolab_question": "TRIG_EVALUATE#\\frac{5}{\\tan(30^{\\circ })}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "5\\sqrt{3}", "decimal": "8.66025…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{5}{\\tan\\left(30^{\\circ\\:}\\right)}=5\\sqrt{3}$$", "input": "\\frac{5}{\\tan\\left(30^{\\circ\\:}\\right)}", "steps": [ { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\tan\\left(30^{\\circ\\:}\\right)=\\frac{\\sqrt{3}}{3}$$", "input": "\\tan\\left(30^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "$$\\tan\\left(x\\right)$$ periodicity table with $$180^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline 30^{\\circ }&\\frac{\\sqrt{3}}{3}\\\\\\hline 45^{\\circ }&1\\\\\\hline 60^{\\circ }&\\sqrt{3}\\\\\\hline 90^{\\circ }&\\pm\\infty\\\\\\hline 120^{\\circ }&-\\sqrt{3}\\\\\\hline 135^{\\circ }&-1\\\\\\hline 150^{\\circ }&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=\\frac{\\sqrt{3}}{3}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=\\frac{5}{\\frac{\\sqrt{3}}{3}}" }, { "type": "interim", "title": "Simplify $$\\frac{5}{\\frac{\\sqrt{3}}{3}}:{\\quad}5\\sqrt{3}$$", "input": "\\frac{5}{\\frac{\\sqrt{3}}{3}}", "result": "=5\\sqrt{3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{\\frac{b}{c}}=\\frac{a\\cdot\\:c}{b}$$", "result": "=\\frac{5\\cdot\\:3}{\\sqrt{3}}" }, { "type": "step", "primary": "Multiply the numbers: $$5\\cdot\\:3=15$$", "result": "=\\frac{15}{\\sqrt{3}}" }, { "type": "interim", "title": "Factor $$15:{\\quad}3\\cdot\\:5$$", "steps": [ { "type": "step", "primary": "Factor $$15=3\\cdot\\:5$$" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{3\\cdot\\:5}{\\sqrt{3}}" }, { "type": "interim", "title": "Cancel $$\\frac{3\\cdot\\:5}{\\sqrt{3}}:{\\quad}\\sqrt{3}\\cdot\\:5$$", "input": "\\frac{3\\cdot\\:5}{\\sqrt{3}}", "result": "=\\sqrt{3}\\cdot\\:5", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "secondary": [ "$$\\sqrt{3}=3^{\\frac{1}{2}}$$" ], "result": "=\\frac{3\\cdot\\:5}{3^{\\frac{1}{2}}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=x^{a-b}$$", "secondary": [ "$$\\frac{3^{1}}{3^{\\frac{1}{2}}}=3^{1-\\frac{1}{2}}$$" ], "result": "=5\\cdot\\:3^{-\\frac{1}{2}+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$1-\\frac{1}{2}=\\frac{1}{2}$$", "result": "=5\\cdot\\:3^{\\frac{1}{2}}" }, { "type": "step", "primary": "Apply radical rule: $$a^{\\frac{1}{n}}=\\sqrt[n]{a}$$", "secondary": [ "$$3^{\\frac{1}{2}}=\\sqrt{3}$$" ], "result": "=5\\sqrt{3}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYuqdzkZru9VBwBVsyZ2b9QWB+Vj7OuQ/OHr8V1BwEwnnq47vuWedXv2WUg94ER8IwX2sjGYkUIdzYuk/DZmjolGBBTEk/JQ2cZ9WKuRzClU7nzcXz/lWIGtnwQRFnThxOedf7z4nMrYu/OLirT7LIz/TRYhLA+B81LSrQYRdZprV" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7KthIVQ/04oD6zagRo8vT+eqrxevU0S7BviPFPoEcZw58kR7hsO/rTOTBE0w4+r1RQslTDKxOR/6J+ZOGvUcauvej3yyjyrovNyFo0tXoh5BkS3dlcCKpQTQcheuut7MkdkqXaZXgCXO2IClMAWKDPTakeLpHgAL5uHHhNWAXFWXqq8Xr1NEuwb4jxT6BHGcOialcV/dI5TH4fXyp+ncwuA==" } } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }