sin(arcsin(1/2)+arctan(-2))

{ "query": { "display": "$$\\sin\\left(\\arcsin\\left(\\frac{1}{2}\\right)+\\arctan\\left(-2\\right)\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\sin(\\arcsin(\\frac{1}{2})+\\arctan(-2))" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "\\frac{\\sqrt{5}}{10}-\\frac{\\sqrt{15}}{5}", "decimal": "-0.55098…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(\\arcsin\\left(\\frac{1}{2}\\right)+\\arctan\\left(-2\\right)\\right)=\\frac{\\sqrt{5}}{10}-\\frac{\\sqrt{15}}{5}$$", "input": "\\sin\\left(\\arcsin\\left(\\frac{1}{2}\\right)+\\arctan\\left(-2\\right)\\right)", "steps": [ { "type": "step", "primary": "Use the following property: $$\\arctan\\left(-x\\right)=-\\arctan\\left(x\\right)$$", "secondary": [ "$$\\arctan\\left(-2\\right)=-\\arctan\\left(2\\right)$$" ], "result": "=\\sin\\left(\\arcsin\\left(\\frac{1}{2}\\right)-\\arctan\\left(2\\right)\\right)" }, { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\sin\\left(\\arcsin\\left(\\frac{1}{2}\\right)\\right)\\cos\\left(\\arctan\\left(2\\right)\\right)-\\cos\\left(\\arcsin\\left(\\frac{1}{2}\\right)\\right)\\sin\\left(\\arctan\\left(2\\right)\\right)$$", "input": "\\sin\\left(\\arcsin\\left(\\frac{1}{2}\\right)-\\arctan\\left(2\\right)\\right)", "result": "=\\sin\\left(\\arcsin\\left(\\frac{1}{2}\\right)\\right)\\cos\\left(\\arctan\\left(2\\right)\\right)-\\cos\\left(\\arcsin\\left(\\frac{1}{2}\\right)\\right)\\sin\\left(\\arctan\\left(2\\right)\\right)", "steps": [ { "type": "step", "primary": "Use the Angle Difference identity: $$\\sin\\left(s-t\\right)=\\sin\\left(s\\right)\\cos\\left(t\\right)-\\cos\\left(s\\right)\\sin\\left(t\\right)$$", "result": "=\\sin\\left(\\arcsin\\left(\\frac{1}{2}\\right)\\right)\\cos\\left(\\arctan\\left(2\\right)\\right)-\\cos\\left(\\arcsin\\left(\\frac{1}{2}\\right)\\right)\\sin\\left(\\arctan\\left(2\\right)\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\sin\\left(\\arcsin\\left(\\frac{1}{2}\\right)\\right)=\\frac{1}{2}$$", "steps": [ { "type": "step", "primary": "Use the following identity: $$\\sin\\left(\\arcsin\\left(x\\right)\\right)=x$$" }, { "type": "step", "result": "=\\frac{1}{2}" } ], "meta": { "interimType": "Trig Use Identity 1Eq" } }, { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\cos\\left(\\arctan\\left(2\\right)\\right)=\\frac{\\sqrt{5}}{5}$$", "input": "\\cos\\left(\\arctan\\left(2\\right)\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\cos\\left(\\arctan\\left(2\\right)\\right)=\\frac{\\sqrt{1+2^{2}}}{1+2^{2}}$$", "steps": [ { "type": "step", "primary": "Use the following identity: $$\\cos\\left(\\arctan\\left(x\\right)\\right)=\\frac{\\sqrt{1+x^{2}}}{1+x^{2}}$$" }, { "type": "step", "result": "=\\frac{\\sqrt{1+2^{2}}}{1+2^{2}}" } ], "meta": { "interimType": "Trig Use Identity 1Eq" } }, { "type": "step", "result": "=\\frac{\\sqrt{1+2^{2}}}{1+2^{2}}" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{\\sqrt{5}}{5}" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\cos\\left(\\arcsin\\left(\\frac{1}{2}\\right)\\right)=\\frac{\\sqrt{3}}{2}$$", "input": "\\cos\\left(\\arcsin\\left(\\frac{1}{2}\\right)\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\cos\\left(\\arcsin\\left(\\frac{1}{2}\\right)\\right)=\\sqrt{1-\\left(\\frac{1}{2}\\right)^{2}}$$", "steps": [ { "type": "step", "primary": "Use the following identity: $$\\cos\\left(\\arcsin\\left(x\\right)\\right)=\\sqrt{1-x^{2}}$$" }, { "type": "step", "result": "=\\sqrt{1-\\left(\\frac{1}{2}\\right)^{2}}" } ], "meta": { "interimType": "Trig Use Identity 1Eq" } }, { "type": "step", "result": "=\\sqrt{1-\\left(\\frac{1}{2}\\right)^{2}}" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{\\sqrt{3}}{2}" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\sin\\left(\\arctan\\left(2\\right)\\right)=\\frac{2\\sqrt{5}}{5}$$", "input": "\\sin\\left(\\arctan\\left(2\\right)\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\sin\\left(\\arctan\\left(2\\right)\\right)=\\frac{2\\sqrt{1+2^{2}}}{1+2^{2}}$$", "steps": [ { "type": "step", "primary": "Use the following identity: $$\\sin\\left(\\arctan\\left(x\\right)\\right)=\\frac{x\\sqrt{1+x^{2}}}{1+x^{2}}$$" }, { "type": "step", "result": "=\\frac{2\\sqrt{1+2^{2}}}{1+2^{2}}" } ], "meta": { "interimType": "Trig Use Identity 1Eq" } }, { "type": "step", "result": "=\\frac{2\\sqrt{1+2^{2}}}{1+2^{2}}" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{2\\sqrt{5}}{5}" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "step", "result": "=\\frac{1}{2}\\cdot\\:\\frac{\\sqrt{5}}{5}-\\frac{\\sqrt{3}}{2}\\cdot\\:\\frac{2\\sqrt{5}}{5}" }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}\\cdot\\:\\frac{\\sqrt{5}}{5}-\\frac{\\sqrt{3}}{2}\\cdot\\:\\frac{2\\sqrt{5}}{5}:{\\quad}\\frac{\\sqrt{5}}{10}-\\frac{\\sqrt{15}}{5}$$", "input": "\\frac{1}{2}\\cdot\\:\\frac{\\sqrt{5}}{5}-\\frac{\\sqrt{3}}{2}\\cdot\\:\\frac{2\\sqrt{5}}{5}", "result": "=\\frac{\\sqrt{5}}{10}-\\frac{\\sqrt{15}}{5}", "steps": [ { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:\\frac{\\sqrt{5}}{5}=\\frac{\\sqrt{5}}{10}$$", "input": "\\frac{1}{2}\\cdot\\:\\frac{\\sqrt{5}}{5}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:\\sqrt{5}}{2\\cdot\\:5}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\sqrt{5}=\\sqrt{5}$$", "result": "=\\frac{\\sqrt{5}}{2\\cdot\\:5}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:5=10$$", "result": "=\\frac{\\sqrt{5}}{10}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8Qmv19wq0Y68zdO7Y2e9yjf7JayTI/yaRHh6qJkHnpeaA585Wz2Y8ioMtXlAhbC3efcFBBxtnU7tj9iPqSMJYMFOYGR0/9M3uEPomWS8obe2Dht/CJXqIwsnmwLWcvt05hjszL1qNmb2/m9oZF4YvIrr2eylJuztmRmE3ucn6ikGqaYBFMq63lqz5op0RunagRImhzl1AluSIs/ALaQHYVk=" } }, { "type": "interim", "title": "$$\\frac{\\sqrt{3}}{2}\\cdot\\:\\frac{2\\sqrt{5}}{5}=\\frac{\\sqrt{15}}{5}$$", "input": "\\frac{\\sqrt{3}}{2}\\cdot\\:\\frac{2\\sqrt{5}}{5}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{\\sqrt{3}\\cdot\\:2\\sqrt{5}}{2\\cdot\\:5}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{\\sqrt{3}\\sqrt{5}}{5}" }, { "type": "interim", "title": "Simplify $$\\sqrt{3}\\sqrt{5}:{\\quad}\\sqrt{15}$$", "input": "\\sqrt{3}\\sqrt{5}", "result": "=\\frac{\\sqrt{15}}{5}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{b}=\\sqrt{a\\cdot{b}}$$", "secondary": [ "$$\\sqrt{3}\\sqrt{5}=\\sqrt{3\\cdot\\:5}$$" ], "result": "=\\sqrt{3\\cdot\\:5}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:5=15$$", "result": "=\\sqrt{15}" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMZDr7b2C7YrlcdHHbaZvc/e0WSaJto/gSg/9oiLU4Wd4X6tVchgKXWOwNTX12hsiAf2i9gqKNBiEkMJvG7+cA4ljl3n2gAnzLZEZd7SUN8+gBhpBW2medQGh/jBx8npQJYmmCeZg/vYI25xQTJ4ImcGtHPT3aYyGho4T+eye3lAK8Jt/6UVH+JlUpVjFDlEQcYyy9OpfQjlc0nN0UxF4lsoeSXGC4MrGtzDC6FpWOH96" } }, { "type": "step", "result": "=\\frac{\\sqrt{5}}{10}-\\frac{\\sqrt{15}}{5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8Qmv19wq0Y68zdO7Y2e9yjc05SS9Mfi1cg/T+Jxx8ll4asHWSnLvgfYYU0cg1ayHTSHFvexuuka7BzMsICzgNy0ZMlruAiU0we9GbKdFB3tOzRqDxPUzBN6vjj5oJL9kUEflbaal4hHa3+SMh28Tayg4C1Zm9GVWxNWr4oOyKFYg2LCCj1mIlbOak8FQ0aA8Yx3iMvWYwwTTugkbawoy4uMeKYFFhGodJZ4mS5Q+F9MLU7jrxhEXFpB5MWqBGpHauy6fwYm+Qmp67A07r8xVv9C22oW6/ETV+JtLfPi4aRe8JyDS30q59Dn8RzHHkwHvjQRmdsnSWQ9j0kMNgvVua4OB57lSmbOxRoix5QonENWJ" } } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }