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Math 1304 Calculus I 3.3 – Derivatives of Trigonometric Functions Trigonometric Functions • Overview Measure Radians, degrees Basic functions sin, cos, tan, csc, sec, cot Periodicity Special values at: 0, π/6, π/4, π/3, π/2, π Sign change Addition formulas Derivatives Angle • Radians: Measure angle by arc length around unit circle θ Definition of Basic Functions hypotenuse opposite θ adjacent • • • • • • sin() = opposite / hypotenuse cos() = adjacent / hypotenuse tan() = opposite / adjacent csc() = hypotenuse / opposite sec() = hypotenuse / adjacent cot() = adjacent / opposite Sin and Cos Give the Others sin( ) tan( ) cos( ) 1 sec( ) cos( ) 1 csc( ) sin( ) cos( ) cot( ) sin( ) Sin, Cos, Tan on Unit Circle tan(θ) θ 1 sin(θ) θ cos(θ) sin( ) tan( ) Periodicity sin( 2 ) sin( ) cos( 2 ) cos( ) tan( 2 ) tan( ) sec( 2 ) sec( ) csc( 2 ) csc( ) cot( 2 ) cot( ) Special Values sin(0) 0 cos(0) 1 1 sin( /6) 2 3 cos(/6) 2 2 cos(/4) 2 1 cos(/3) 2 cos(/2) 0 . 2 sin( /4) 2 3 sin( /3) 2 sin( /2) 1 . Basic Inequalities For tan(θ) θ 1 sin(θ) θ cos(θ) 0 / 2 sin( ) tan( ) Proof of Basic Equalities D sin( ) BC BA sin( ) B tan(θ) Draw tangent line at B. It intersects AD at E θ 1 sin(θ) O E θ cos(θ) C A BE EA DE EA tan( ) Special Limit lim 0 sin( ) 1 Use Squeezing Theorem sin( ) tan( ) sin( ) / sin( ) / sin( ) tan( ) / sin( ) 1 / sin( ) 1 / cos( ) cos( ) sin( ) / 1 lim cos( ) lim 0 0 1 lim 0 sin( ) sin( ) lim 1 0 1 lim 0 sin( ) 1 Another Special Limit lim 0 1 cos( ) 0 Addition Formulas • sin(x+y) = sin(x) cos(y) + cos(x) sin(y) • cos(x+y) = cos(x) cos(y) – sin(x) sin(y) Derivative of Sin and Cos • Use addition formulas (in class) Derivatives • • • • • • If f(x) = sin(x), then f’(x) = cos(x) If f(x) = cos(x), then f’(x) = - sin(x) If f(x) = tan(x), then f’(x) = sec2(x) If f(x) = csc(x), then f’(x) = - csc(x) cot(x) If f(x) = sec(x), then f’(x) = sec(x) tan(x) If f(x) = cot(x), then f’(x) = - csc2(x) A good working set of rules • • • • • • • • • • • • Constants: If f(x) = c, then f’(x) = 0 Powers: If f(x) = xn, then f’(x) = nxn-1 Exponentials: If f(x) = ax, then f’(x) = (ln a) ax Trigonometric Functions: If f(x) = sin(x), then f’(x)=cos(x) If f(x) = cos(x), then f’(x) = -sin(x) If f(x)= tan(x), then f’(x) = sec2(x) If f(x) = csc(x), then f’(x) = -csc(x) cot(x) If f(x)= sec(x), then f’(x) = sec(x)tan(x) If f(x) = cot(x), then f’(x) = -csc2(x) Scalar mult: If f(x) = c g(x), then f’(x) = c g’(x) Sum: If f(x) = g(x) + h(x), then f’(x) = g’(x) + h’(x) Difference: If f(x) = g(x) - h(x), then f’(x) = g’(x) - h’(x) Multiple sums: derivative of sum is sum of derivatives Linear combinations: derivative of linear combo is linear combo of derivatives Product: If f(x) = g(x) h(x), then f’(x) = g’(x) h(x) + g(x)h’(x) Multiple products: If F(x) = f(x) g(x) h(x), then F’(x) = f’(x) g(x) h(x) + … Quotient: If f(x) = g(x)/h(x), then f’(x) = (g’(x) h(x) - g(x)h’(x))/(h(x))2