Math 1304 Calculus I

1. Math 1304 Calculus I 3.2 – Derivatives of Trigonometric Functions

2. Trigonometric Functions • 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