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Understanding Derivatives of Trigonometric Functions in Calculus

This overview of derivatives of trigonometric functions provides key definitions and formulas essential for calculus. It explores the measurement of angles in radians, special values at common angles, and the periodic nature of functions like sin, cos, and tan. The document outlines addition formulas for sine and cosine, derivative rules for trigonometric functions, and general rules for constants, powers, and products. By mastering these concepts, students can simplify calculus applications involving trigonometric functions and their derivatives effectively.

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Understanding Derivatives of Trigonometric Functions in Calculus

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  1. Math 1304 Calculus I 3.3 – Derivatives of Trigonometric Functions

  2. 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

  3. Angle • Radians: Measure angle by arc length around unit circle θ

  4. Definition of Basic Functions hypotenuse • sin() = opposite / hypotenuse • cos() = adjacent / hypotenuse • tan() = opposite / adjacent • csc() = hypotenuse / opposite • sec() = hypotenuse / adjacent • cot() = adjacent / opposite opposite θ adjacent

  5. Sin and Cos Give the Others

  6. Sin, Cos, Tan on Unit Circle tan(θ) θ 1 sin(θ) θ cos(θ)

  7. Periodicity

  8. Special Values

  9. Basic Inequalities For tan(θ) θ 1 sin(θ) θ cos(θ)

  10. Proof of Basic Equalities D B tan(θ) Draw tangent line at B. It intersects AD at E θ 1 E sin(θ) θ O A C cos(θ)

  11. Special Limit

  12. Use Squeezing Theorem

  13. Another Special Limit

  14. Addition Formulas • sin(x+y) = sin(x) cos(y) + cos(x) sin(y) • cos(x+y) = cos(x) cos(y) – sin(x) sin(y)

  15. Derivative of Sin and Cos • Use addition formulas (in class)

  16. 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)

  17. 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

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