Lesson 3 (08/28/26)

Original Notes

Page 1

Lesson 3 (08/28/26)

Today:

  • Measuring Angle
  • Review of Trig. Functions
  • Inverse Trig. Functions
  • Examples

Office Hours: MWF 1:30pm – 2:30pm, MATH 842

Announcements:

  • MRR opens on Monday (MATH G175)
  • Quiz 1 on Tuesday (Lesson 1, 2)

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Warmup:

Right-angled triangle diagram defining the trigonometric ratios for an angle theta at the bottom-left vertex. The base adjacent to theta is labeled 'adj.', the vertical side opposite theta is labeled 'Opp.', and the longest slanted side opposite the right angle is labeled 'Hyp.' (hypotenuse).
Visual Description: Right-angled triangle diagram defining the trigonometric ratios for an angle theta at the bottom-left vertex. The base adjacent to theta is labeled 'adj.', the vertical side opposite theta is labeled 'Opp.', and the longest slanted side opposite the right angle is labeled 'Hyp.' (hypotenuse).

\[ \sin\theta = \frac{\text{opp.}}{\text{hyp.}}, \qquad \operatorname{cosec}\theta = \frac{1}{\sin\theta} \]

\[ \cos\theta = \frac{\text{adj.}}{\text{hyp.}}, \qquad \sec\theta = \frac{1}{\cos\theta} \]

\[ \tan\theta = \frac{\text{opp.}}{\text{adj.}} = \frac{\sin\theta}{\cos\theta}, \qquad \cot\theta = \frac{1}{\tan\theta} \]

Reciprocal

these are all periodic function

there is a \(p\) such that \(f(x+p) = f(x)\)

Page 3

Unit circle in the Cartesian coordinate plane centered at the origin (0,0) showing points (0,1) and (1,0). A point (x,y) lies on the circle in the first quadrant, connected to the origin with a hypotenuse of length 1 making an angle theta with the positive x-axis. A right-angled triangle is drawn with the adjacent side along the horizontal axis labeled adj = x and the opposite vertical side labeled opp = y.
Visual Description: Unit circle in the Cartesian coordinate plane centered at the origin (0,0) showing points (0,1) and (1,0). A point (x,y) lies on the circle in the first quadrant, connected to the origin with a hypotenuse of length 1 making an angle theta with the positive x-axis. A right-angled triangle is drawn with the adjacent side along the horizontal axis labeled adj = x and the opposite vertical side labeled opp = y.

\(\cos\theta = x\) is Horizontal distance

  • +ve in \(\text{I}\) & \(\text{IV}\) quadrant
  • -ve in \(\text{II}\) & \(\text{III}\) quadrant

\(\sin\theta = y\) is Vertical distance

  • +ve in \(\text{I}\) & \(\text{II}\) quadrant
  • -ve in \(\text{III}\) & \(\text{IV}\)

\(\theta\) is +ve if measured \(\text{CCW}\) w.r.t +ve x-axis

-ve if measured clock wise

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Measuring Angle

\[ 1\text{ full circle} = 360\text{ Degrees} = 2\pi\text{ Radian} \]

Unless specified, angles in calculus always measured in Radians.

A circle showing a central sector with radius \(r\), central angle \(\theta\), and subtended arc length \(s\) highlighted along the circumference. Adjacent text defines \(s = \text{Arc length}\) and the arc length formula \(s = r\theta\).
Visual Description: A circle showing a central sector with radius \(r\), central angle \(\theta\), and subtended arc length \(s\) highlighted along the circumference. Adjacent text defines \(s = \text{Arc length}\) and the arc length formula \(s = r\theta\).

\[ 1\text{ Radian} = \text{Angle measure when radius equals Arc length.} \]

Conversion:

\[ 1^\circ = \frac{2\pi}{360}\text{ Radian} \] \[ 1\text{ Radian} = \frac{360^\circ}{2\pi} \]

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\(\sin\theta\):

Graph of the sine function y = sin(theta) plotted on Cartesian axes. The wave oscillates periodically with a peak labeled by a red vertical double arrow indicating the amplitude of 1. The horizontal axis marks 0 at the origin, the first positive half-cycle ending at pi, and a full cycle ending at 2pi, indicated by a horizontal purple arrow showing a period of 2pi. Annotations in the top-right corner state 'Amplitude: 1' and 'Period = 2pi'.
Visual Description: Graph of the sine function y = sin(theta) plotted on Cartesian axes. The wave oscillates periodically with a peak labeled by a red vertical double arrow indicating the amplitude of 1. The horizontal axis marks 0 at the origin, the first positive half-cycle ending at pi, and a full cycle ending at 2pi, indicated by a horizontal purple arrow showing a period of 2pi. Annotations in the top-right corner state 'Amplitude: 1' and 'Period = 2pi'.

Transformation:

\[ f(\theta) = a \sin(b(\theta - c)) + d \]

  • \(a\): Stretches vertically; \(\text{Amplitude} = a\)
  • \(b\): Stretches horizontally; \(\text{Period} = \frac{2\pi}{b}\)
  • \(c\): Shift horizontally
  • \(d\): Shift vertically

Page 6

Inverse of \(\sin\theta\)

A graph of the function sin(theta) along horizontal theta-axis and vertical axis. The blue sine wave oscillates across multiple periods with a horizontal test line drawn across it showing multiple intersections. A single period from -pi/2 to pi/2 is highlighted in magenta/pink to indicate the restricted domain. Key horizontal points are marked including pi and 2pi with a horizontal purple arrow. The curve is labeled sin(theta).
Visual Description: A graph of the function sin(theta) along horizontal theta-axis and vertical axis. The blue sine wave oscillates across multiple periods with a horizontal test line drawn across it showing multiple intersections. A single period from -pi/2 to pi/2 is highlighted in magenta/pink to indicate the restricted domain. Key horizontal points are marked including pi and 2pi with a horizontal purple arrow. The curve is labeled sin(theta).

Not invertible on \((-\infty, \infty)\)

Restrict the Domain to \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \leadsto\) it is invertible

\[ \arcsin(x) = \sin^{-1}(x) = \theta \leadsto \text{then } \theta \text{ lies in } \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \text{ such that } \sin\theta = x \]

\(D: [-1, 1]\)

\(R: \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)

Graph of the inverse sine function, labeled sin^{-1}(x). The horizontal x-axis ranges from -1 to 1, and the vertical y-axis ranges from -pi/2 to pi/2. The curve is an increasing S-shape passing through the origin (0, 0), with end points at (-1, -pi/2) and (1, pi/2).
Visual Description: Graph of the inverse sine function, labeled sin^{-1}(x). The horizontal x-axis ranges from -1 to 1, and the vertical y-axis ranges from -pi/2 to pi/2. The curve is an increasing S-shape passing through the origin (0, 0), with end points at (-1, -pi/2) and (1, pi/2).

Page 7

eg: Evaluate \(\sin^{-1}\left(\frac{1}{2}\right)\)

Find \(\theta\) in \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) such that \(\sin\theta = \frac{1}{2}\)

Hand-drawn unit circle diagram on a coordinate plane illustrating angles in the first quadrant with coordinate points: (1, 0) on the positive x-axis, (\sqrt{3}/2, 1/2) for \pi/6, (\sqrt{2}/2, \sqrt{2}/2) for \pi/4, (1/2, \sqrt{3}/2) for \pi/3, and (0, 1) on the positive y-axis.
Visual Description: Hand-drawn unit circle diagram on a coordinate plane illustrating angles in the first quadrant with coordinate points: (1, 0) on the positive x-axis, (\sqrt{3}/2, 1/2) for \pi/6, (\sqrt{2}/2, \sqrt{2}/2) for \pi/4, (1/2, \sqrt{3}/2) for \pi/3, and (0, 1) on the positive y-axis.

\(\left[-\frac{\pi}{2}, 0\right]\) is IV Q

\(\left[0, \frac{\pi}{2}\right]\) is I Q

want \(\theta\) in \(\left[0, \frac{\pi}{2}\right]\) such that \(\sin\theta = \frac{1}{2}\)

\[\theta = \pi/6\]

NOT \(5\pi/6\), bcz its not in \(\left[0, \frac{\pi}{2}\right] = \text{Q}_1\)

Page 8

Inverse of \(\cos\theta\)

Graph of the periodic function y = \cos\theta with a horizontal line drawn across it illustrating the failure of the horizontal line test on (-\infty, \infty). A restricted interval from \theta = 0 to \theta = \pi is highlighted in magenta, showing the portion of the curve where cosine is one-to-one, decreasing monotonically from 1 to -1. The x-axis indicates key points at \pi and 2\pi, and the curve is labeled \cos\theta.
Visual Description: Graph of the periodic function y = \cos\theta with a horizontal line drawn across it illustrating the failure of the horizontal line test on (-\infty, \infty). A restricted interval from \theta = 0 to \theta = \pi is highlighted in magenta, showing the portion of the curve where cosine is one-to-one, decreasing monotonically from 1 to -1. The x-axis indicates key points at \pi and 2\pi, and the curve is labeled \cos\theta.

Not invertible on \((-\infty, \infty)\)

Restrict Domain to \([0, \pi]\) we can define \(\arccos(x) = \cos^{-1}(x)\)

\[ \arccos(x) = \cos^{-1}(x) = \theta \iff \text{then } \cos\theta = x, \quad \theta \underset{\text{is in}}{\in} [0, \pi] \]

\(\text{D: } [-1, 1]\)
\(\text{R: } [0, \pi]\)

Graph of the inverse cosine function y = \cos^{-1}(x). The horizontal axis (domain) ranges from -1 to 1, and the vertical axis (range) ranges from 0 to \pi, with a labeled tick mark at \frac{\pi}{2}. The curve decreases continuously from (-1, \pi) through the y-intercept at (0, \frac{\pi}{2}) down to the x-intercept at (1, 0), labeled \cos^{-1}(x).
Visual Description: Graph of the inverse cosine function y = \cos^{-1}(x). The horizontal axis (domain) ranges from -1 to 1, and the vertical axis (range) ranges from 0 to \pi, with a labeled tick mark at \frac{\pi}{2}. The curve decreases continuously from (-1, \pi) through the y-intercept at (0, \frac{\pi}{2}) down to the x-intercept at (1, 0), labeled \cos^{-1}(x).

Page 9

eg: Evaluate \(\cos^{-1}\left(\cos\left(\frac{4\pi}{3}\right)\right)\)

\[\neq \frac{4\pi}{3}\]

Find \(\theta\) in \([0, \pi]\) such that

\[\cos^{-1}\left(\cos\left(\frac{4\pi}{3}\right)\right) = \theta\] \[\cos\theta = \cos\left(\frac{4\pi}{3}\right)\]

Unit circle centered on the Cartesian plane illustrating angle values with reference angles of pi/3. Lines extend from the origin into all four quadrants at angles pi/3, 2pi/3, 4pi/3, and 5pi/3. Red curved arrows indicate angles starting from the positive x-axis: one measuring pi/3 in the first quadrant, another measuring 2pi/3 in the second quadrant, and a third extending around into the third quadrant to 4pi/3, demonstrating the cosine symmetry between 2pi/3 and 4pi/3.
Visual Description: Unit circle centered on the Cartesian plane illustrating angle values with reference angles of pi/3. Lines extend from the origin into all four quadrants at angles pi/3, 2pi/3, 4pi/3, and 5pi/3. Red curved arrows indicate angles starting from the positive x-axis: one measuring pi/3 in the first quadrant, another measuring 2pi/3 in the second quadrant, and a third extending around into the third quadrant to 4pi/3, demonstrating the cosine symmetry between 2pi/3 and 4pi/3.

\[\cos\left(\frac{2\pi}{3}\right) = \cos\left(\frac{4\pi}{3}\right)\] \[\cos^{-1}\left(\cos\left(\frac{4\pi}{3}\right)\right) = \frac{2\pi}{3}\]

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A unit circle on a Cartesian coordinate system illustrating trigonometric symmetry properties. A rectangle inscribed within the circle connects four symmetric points: \( (x, y) \) in Quadrant I at angle \( \theta \), \( (-x, y) \) in Quadrant II at angle \( \pi - \theta \), \( (-x, -y) \) in Quadrant III at angle \( \pi + \theta \), and \( (x, -y) \) in Quadrant IV at angle \( -\theta \). The horizontal component \( x \) and vertical component \( y \) are indicated for the right triangle in Quadrant I, with red directional arcs illustrating the angle rotations \( \theta \), \( -\theta \), \( \pi - \theta \), and \( \pi + \theta \) from the positive x-axis.
Visual Description: A unit circle on a Cartesian coordinate system illustrating trigonometric symmetry properties. A rectangle inscribed within the circle connects four symmetric points: \( (x, y) \) in Quadrant I at angle \( \theta \), \( (-x, y) \) in Quadrant II at angle \( \pi - \theta \), \( (-x, -y) \) in Quadrant III at angle \( \pi + \theta \), and \( (x, -y) \) in Quadrant IV at angle \( -\theta \). The horizontal component \( x \) and vertical component \( y \) are indicated for the right triangle in Quadrant I, with red directional arcs illustrating the angle rotations \( \theta \), \( -\theta \), \( \pi - \theta \), and \( \pi + \theta \) from the positive x-axis.

\[ \cos \theta = x \]

\[ \cos(-\theta) = x \]

\[ \cos(\pi - \theta) = \cos(\pi + \theta) \]

Page 11

Inverse of \(\tan\theta\)

Graph of y = tan(theta) plotted on Cartesian axes with vertical dashed asymptotes at theta = -pi/2 and theta = pi/2. Multiple periodic branches are drawn, with the central branch passing through the origin (0,0) highlighted in pink/purple to represent the restricted domain (-pi/2, pi/2).
Visual Description: Graph of y = tan(theta) plotted on Cartesian axes with vertical dashed asymptotes at theta = -pi/2 and theta = pi/2. Multiple periodic branches are drawn, with the central branch passing through the origin (0,0) highlighted in pink/purple to represent the restricted domain (-pi/2, pi/2).

Restrict \(D: \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), \(R: (-\infty, \infty) \leadsto \tan x\) is invertible

\[ \arctan(x) = \tan^{-1}(x) = \theta \leadsto \tan\theta = x, \quad \theta \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \]

\(D: (-\infty, \infty)\)
\(R: \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

Graph of y = tan^(-1)(x) (or arctan(x)) on Cartesian axes with horizontal dashed asymptotes at y = pi/2 and y = -pi/2. The function is strictly increasing, passing through the origin (0,0), approaching y = pi/2 as x goes to positive infinity, and approaching y = -pi/2 as x goes to negative infinity. Labelled tan^(-1)(x).
Visual Description: Graph of y = tan^(-1)(x) (or arctan(x)) on Cartesian axes with horizontal dashed asymptotes at y = pi/2 and y = -pi/2. The function is strictly increasing, passing through the origin (0,0), approaching y = pi/2 as x goes to positive infinity, and approaching y = -pi/2 as x goes to negative infinity. Labelled tan^(-1)(x).