tan((7pi)/4)

{ "query": { "display": "$$\\tan\\left(\\frac{7π}{4}\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\tan(\\frac{7π}{4})" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "-1", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\tan\\left(\\frac{7π}{4}\\right)=-1$$", "input": "\\tan\\left(\\frac{7π}{4}\\right)", "steps": [ { "type": "interim", "title": "$$\\tan\\left(\\frac{7π}{4}\\right)=\\tan\\left(\\frac{3π}{4}\\right)$$", "input": "\\tan\\left(\\frac{7π}{4}\\right)", "result": "=\\tan\\left(\\frac{3π}{4}\\right)", "steps": [ { "type": "step", "primary": "Rewrite $$\\frac{7π}{4}$$ as $$π+\\frac{3π}{4}$$", "result": "=\\tan\\left(π+\\frac{3π}{4}\\right)" }, { "type": "step", "primary": "Apply the periodicity of $$\\tan$$: $$\\tan\\left(x+π\\right)=\\tan\\left(x\\right)$$", "secondary": [ "$$\\tan\\left(π+\\frac{3π}{4}\\right)=\\tan\\left(\\frac{3π}{4}\\right)$$" ], "result": "=\\tan\\left(\\frac{3π}{4}\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\frac{\\sin\\left(\\frac{3π}{4}\\right)}{\\cos\\left(\\frac{3π}{4}\\right)}$$", "input": "\\tan\\left(\\frac{3π}{4}\\right)", "result": "=\\frac{\\sin\\left(\\frac{3π}{4}\\right)}{\\cos\\left(\\frac{3π}{4}\\right)}", "steps": [ { "type": "step", "primary": "Use the basic trigonometric identity: $$\\tan\\left(x\\right)=\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}$$", "result": "=\\frac{\\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:$${\\quad}\\sin\\left(\\frac{3π}{4}\\right)=\\frac{\\sqrt{2}}{2}$$", "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{\\frac{\\sqrt{2}}{2}}{-\\frac{\\sqrt{2}}{2}}" }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{\\sqrt{2}}{2}}{-\\frac{\\sqrt{2}}{2}}:{\\quad}-1$$", "input": "\\frac{\\frac{\\sqrt{2}}{2}}{-\\frac{\\sqrt{2}}{2}}", "result": "=-1", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\frac{\\sqrt{2}}{2}}{\\frac{\\sqrt{2}}{2}}" }, { "type": "step", "primary": "Divide fractions: $$\\frac{\\frac{a}{b}}{\\frac{c}{d}}=\\frac{a\\cdot\\:d}{b\\cdot\\:c}$$", "result": "=-\\frac{\\sqrt{2}\\cdot\\:2}{2\\sqrt{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$\\sqrt{2}$$", "result": "=-\\frac{2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajbJ3wo1uFgyKliORhBgbJo0ussnnnisJndCAm4z+zno8qSLiRo0DO9IhVuX5Gm3ZK3CQoYlYQ8U+Tfyx0kyzI8hBup7XxSVaYGLLOwF6kQecmNwEgvmmajzdQwVlFLMmxS3xXPiuLkj+jJktEBj2/1IAQNpETCgQ7A7uB2HlXgboNExR2KxRxAUdkyDIsIWVgNpK7SU09JD4Nr69pNDewic=" } } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }