Section Derivatives of the Trigonometric Functions
When a mathematician says something is “true,” s/he means that s/he can prove it from a set of axioms and definitions. This is called deductive reasoning. We used deductive reasoning on Problem 2.25 (product rule) and on Problem 2.26 (quotient rule) by proving (deducing) them from the definition of the derivative. When a scientist says something is true, s/he means s/he believes it to be true based on data. This is called inductive reasoning because s/he induces the truth from observations. We are going to determine the derivative of \(s(x) = \sin(x)\) in two different ways, inductively and deductively.
Problem 2.40.
Inductive: Use the unit circle to fill in the second row of Table 2.41 and make a very accurate graph of the sine function using this data, labeling each of these points on the graph. Sketch the tangent line to the graph of sine at each of these points.
| \(x\) | 0 | \(\pi/4\) | \(\pi/2\) | \(3\pi/4\) | \(\pi\) | \(5\pi/4\) | \(3\pi/2\) | \(7\pi/4\) | \(2\pi\) |
| \(s(x)\) | |||||||||
| \(s'(x)\) | |||||||||
Problem 2.42.
Approximate the slope of each tangent line you graphed in the last problem and fill in the third row of Table 2.41 with this data. Sketch this new function, the derivative of sine, by plotting the points \((x,s'(x))\) from the table. Does the graph of \(s'\) look familiar?
Based on the last two problems, we have a guess for the derivative of sine. Now, we'll see if our guess is correct.
Trigonometry Reminder — The Sum and Difference Identities. The appendix Trigonmetry has common identities you will need to know, but ask and I will give a quick review.
\(\sin(x \pm y) = \sin(x)\cos(y) \pm \cos(x)\sin(y)\)
\(\cos(x \pm y) = \cos(x)\cos(y) \mp \sin(x)\sin(y)\)
Problem 2.43.
Deductive: Compute the derivative of the sine function using Definition 2.1 and the Trigonometry Reminder above.
Problem 2.44.
Compute the derivative of \(f(x) = \cos(x).\)
Problem 2.45.
Compute the derivatives of the tangent, cotangent, secant, and cosecant functions by using the derivatives of sine and cosine along with the quotient rule.
Problem 2.46.
Find a function f with derivative \(f'(x) = 3\cos(x) - 2\sin(x) + x + 2.\)
Problem 2.47.
If \(\dsp y = \frac{\cos(x)}{x+3}\text{,}\) compute \(y'\text{.}\)
Algebra Reminder. Any point where a function crosses the x-axis is called an x-intercept of the function, and any point where a function crosses the y-axis is called a y-intercept of the function. To find an x-intercept we set \(y\) equal to zero and solve for \(x.\) To find a y-intercept we set \(x\) equal to zero and solve for \(y.\)
Problem 2.48.
Find numbers a, b, and c so that the function \(f(x) = ax^2 + bx + c\) has an x-intercept of 2, a y- intercept of 2, and a tangent line with slope of 2 at \((2,4a+2b+c).\)
Problem 2.49.
If \(y = \sin(x)\cos(x),\) compute \(y'\text{.}\)
Problem 2.50.
Suppose that \(L\) is a line tangent to \(y=\cos(x)\) at the point, \((c,f(c))\) and \(L\) passes through the point \((0,0).\) Show that \(c\) satisfies the equation \(t = -\cot(t).\) That is show that \(c = -\cot(c)\text{.}\)
Problem 2.51.
Find the equation of the line tangent to the curve \(y = \tan(x) + \pi\) at the point on the curve where \(x = \pi.\) Repeat this exercise for \(x = \pi/4\text{.}\)