Math 140 Week 1 Notes

Functions and Geometric Objects

• Course begins with simple geometric objects within the framework of functions.
• A function is a fundamental tool in mathematics.
• It's a rule (implicit or explicit) that governs the relationship between inputs (x-values) and outputs (y-values).
• Simplest scenario: one input and one output.
• Expressed as $y = f(x)$, read as "y equals f of x".
• 'x' is the input, 'y' is the output, 'f' is the rule or the function itself.
• When inputs and outputs are numbers, they can be visualized as a graph.
• Example: Graph the function $y = 2x + 1$.

Example: Graphing a Function

• Example: Graph the function $y = \sqrt{7-x^2}$.
• Solution: Semi-circle.
• These ideas should already be familiar.

Straight Lines

• Start with a straight line.
• Key aspects of straight lines:
  • Equation: $y = mx + b$
  • Graphs: Continuous straight line, stretches infinitely on both sides.
  • Slopes: Can be positive or negative, determine steepness and distance between points.
  • Intercepts: Has x and y intercepts.
    • $y = a$ has a slope of 0.
    • $x = b$ has an undefined slope.
    • $y = a$ has only one y-intercept.
    • $x = b$ has only one x-intercept.

Relationships Between Two Straight Lines

• Consider two straight lines and their possible relationships.
• Three distinct possibilities:
  • Case 1: Parallel lines
  • Case 2: Intersecting lines
  • Case 3: Collinear lines
• These cases should be previously explored.

Relationships Between Three Straight Lines

• What about the case with 3 lines?
• Bonus: How many distinct drawings can be made with 3 lines?
• A particular case: 3 lines enclose an area.
• The figure thus formed is called a Triangle.

Triangles

• Major focus in the first week: Triangles.
• Study of triangles is important throughout history because it is the simplest closed object.
• All complex objects made out of straight lines can be broken down into triangles.
• The breakdown may not be unique.
• Triangle:
  • Corners are labeled with capital letters (A, B, C).
  • Sides are labeled with small letters (a, b, c).
  • 6 fundamental quantities: 3 sides and 3 angles.

Relationships Between Sides and Angles of Triangles

• Discover relationships between the 6 quantities (3 sides, 3 angles).
• Dealing with measurements, so we need standards.
• Measuring lengths (sides of triangles) is familiar.
• Measuring angles needs more discussion.
• What is an angle?
• To talk about angles, we need two straight lines.
• The angle between two lines is a measure of how much you need to rotate one of the lines so that it matches up exactly with the other line.
• There can be more than one answer (more than one angle).

Measuring Angles

• Since we use the term "rotation", think of the simplest object that associates with rotation: a circle.
• Measuring angles in degrees.
• Arbitrarily define 1 full rotation of a circle to equal 360 degrees, written as $360^\circ$.
• Due to it being an arbitrary number, this is NOT a natural way to measure angles. A much more natural measurement of angles will be covered later on: radian measure.

Activity: Exploring Triangle Angles

• Draw any triangle.
• Cut it out.
• Using it as a reference (tracing etc.), cut out two more copies.
• Now you have 3 total copies of the same triangle.
• Label the angles as 1, 2, and 3, making sure that the labels on all 3 triangles agree.
• Arrange the triangles so that they stack.
• Observe what happens.
• You get two straight lines.
• You have all three angles 1, 2, and 3.

The Sum of Angles in a Triangle

• No matter what triangle you start with, the three triangle's bases align to a straight line.
• A straight line corresponds to a half-rotation which in turn corresponds to $180^\circ$.
• In a triangle, the sum of three angles is always $180^\circ$.
• This is a fact you've already known.
• One relationship between angles of a triangle:
  • $\angle A + \angle B + \angle C = 180^\circ$
• There are infinitely many possible combinations of values of $\angle A$, $\angle B$, and $\angle C$ that satisfy the given relationship.

Right Triangles

• Simplify things by focusing on triangles where at least one of the angles is exactly $90^\circ$.
• Triangles which have their angles equal to $90^\circ$ are called Right-triangles.
• Considered "simpler objects" because they only have exactly 1 angle that is to $90^\circ$.
• Any triangle can be broken down into 2 smaller right triangles.
• This justifies our claim that the simplest object we can begin our study with is a right-angled triangle.

Magic of Angles

• Recall: Need 2 lines to talk about angles, and need 3 lines to talk about triangles, so that angle is independent of the length of sides of the triangle.
• Triangles can have the same angle but have different side lengths.
• Basis of Trigonometry.

Relationships Between Angles and Sides

• In our quest for a relationship between angles and sides of a triangle, we have hit a roadblock.
• There seems to be no direct relationships.
• Thus, we turn our attention to lengths of the sides.
• Natural consideration: relationships between sides, not the actual lengths.

Ratio Between Sides

• Since an angle represents a relationship between 2 sides, in order to relate it with side-lengths, it is natural to expect something if relationships between 2 side-lengths are considered.
• Since side-lengths are just numbers, the primary candidate for what this relationship might be is the RATIO between the sides.
• Ratio between 2 numbers a and b is essentially the rational number represented as $\frac{a}{b}$, written as $a:b$.

Activity: Exploring Ratios of Sides

• Draw two lines.
• Pick 3 points (A, B, C) on the horizontal line.
• Using a ruler, draw vertical lines from B, C, and D such that it meets the other line at E, F, and G respectively.
• Measure the lengths of the 3 sides of triangles $\triangle AEB$, $\triangle AFC$, and $\triangle AGD$.

Observations and Calculations

• Example measurements (these will vary):
  • $\triangle AEB$: AE = 4, EB = 1.4, AB = 4.2
  • $\triangle AFC$: AF = 5.4, FC = 1.7, AC = 5.2
  • $\triangle AGD$: AG = 6.6, GD = 2.1, AD = 6.2
• Now for each $\triangle AEB$, calculate these three ratios:
  • $\frac{AE}{AB} = \frac{4}{4.2} = 0.95$
  • $\frac{AE}{EB} = \frac{4}{1.4} = 2.86$
  • $\frac{EB}{AB} = \frac{1.4}{4.2} = 0.33$
• For $\triangle AFC$:
  • $\frac{AF}{AC} = \frac{5.4}{5.2} = 1.04$
  • $\frac{AF}{FC} = \frac{5.4}{1.7} = 3.18$
  • $\frac{FC}{AC} = \frac{1.7}{5.2} = 0.33$
• For $\triangle AGD$:
  • $\frac{AG}{AD} = \frac{6.6}{6.2} = 1.06$
  • $\frac{AG}{GD} = \frac{6.6}{2.1} = 3.14$
  • $\frac{GD}{AD} = \frac{2.1}{6.2} = 0.34$

Trigonometric Ratios

• The fact that the ratios line up is because of our prior observation that the angle ($\angle A$) is shared between all the triangles.
• This tells us that there is some relationship between the ratio of sides of a triangle and a particular angle.
• This is the point where we embark upon this study, called the study of trigonometric ratios.
• Given a triangle $\triangle ABC$ (right-angled):
  • There are 6 possible ratios that can be formed using 3 side lengths:
    • $\frac{AC}{AB}$
    • $\frac{BC}{AB}$
    • $\frac{BC}{AC}$
    • $\frac{AB}{AC}$
    • $\frac{AB}{BC}$
    • $\frac{AC}{BC}$
  • Ratios in each block are reciprocals, and they are assigned six different names:
    • $\frac{AC}{AB}$: Sine of angle $\theta$ or $\sin \theta$
    • $\frac{BC}{AB}$: Cosine of angle $\theta$ or $\\cos \theta$
    • $\frac{BC}{AC}$: Tangent of angle $\theta$ or $\tan \theta$
    • $\frac{AB}{BC}$: Cosecant of angle $\theta$ or $\csc \theta$
    • $\frac{AC}{BC}$: Secant of angle $\theta$ or $\sec \theta$
    • $\frac{AB}{AC}$: Cotangent of angle $\theta$ or $\cot \theta$

Mnemonics for Trigonometric Ratios

• The ratios depend on which angle '$\theta$' is under consideration.
• A Mnemonic:
  • SOH CAH TOA
    • Sin: Opposite / Hypotenuse
    • Cos: Adjacent / Hypotenuse
    • Tan: Opposite / Adjacent
• Hypotenuse never changes.

Examples of Finding Trigonometric Ratios

• Example: Given a right triangle with sides 3 (adjacent), 4 (opposite), and 5 (hypotenuse), find the six ratios:
  • $\sin \theta = \frac{4}{5}$
  • $\cos \theta = \frac{3}{5}$
  • $\tan \theta = \frac{4}{3}$
  • $\csc \theta = \frac{5}{4}$
  • $\sec \theta = \frac{5}{3}$
  • $\cot \theta = \frac{3}{4}$
• Example: Given a right triangle with sides 1 (opposite), 1 (adjacent), and $\sqrt{2}$ (hypotenuse), find the six ratios:
  • $\sin \theta = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
  • $\cos \theta = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
  • $\tan \theta = 1$
  • $\csc \theta = \sqrt{2}$
  • $\sec \theta = \sqrt{2}$
  • $\cot \theta = 1$

Pythagorean Theorem and Trigonometric Ratios

• An additional tool/relationship we have between sides of a right-angled triangle is the Pythagorean theorem, namely
• "Sum of squares of the legs equals the square of the hypotenuse"
• Or $\text{AC}^2 = \text{AB}^2 + \text{BC}^2$
• Example: Find AC if AB = 2 and BC = 1:
  • $\text{AC}^2 = (1)^2 + (2)^2$
  • $\text{AC}^2 = 1 + 4 = 5$
  • $\text{AC} = \sqrt{5}$

Further Examples

*Example: Find the six trigonometric ratios of an angle in a triangle with adjacent side 2, opposite side 3, and hypotenuse $\sqrt{13}$:
* $\sin \theta = \frac{3}{\sqrt{13}}$
* $\cos \theta = \frac{2}{\sqrt{13}}$
* $\tan \theta = \frac{3}{2}$
* $\csc \theta = \frac{\sqrt{13}}{3}$
* $\sec \theta = \frac{\sqrt{13}}{2}$
* $\cot \theta = \frac{2}{3}$

• The adjacent side is 2 and the opposite is 3. The hypotenuse is $\sqrt{13}$
  $\text{AC}^2 = (3)^2 + (2)^2 = 9 + 4 = 13$
  $\text{AC} = \sqrt{13}$

Application: Finding Height of a Building

• It uses trigonometric ratios to find the height of a building without directly measuring it.
• Example: How tall is the tower? How high is the window?
• The distance is 325ft and the angles are 39 and 25.
  $\tan(39) = \frac{x}{325}$
  $\tan(39) \times 325ft = x$
  $x = 263.18$
• The total height is 263.18ft.
  $\tan(25) = \frac{x}{325}$
  $\tan(25) \times 325ft = x$
  $x = 151.55$
• The height of the window is 151.55 ft.

Radian Measure

• Develop a more natural measuring unit for angles.
• It is not quite arbitrary, relating angles to circles.
• Angles relate to rotation, which relates to circles.

Circles and Angles

• Concentric circles have the same center.
• Even though lengths AC, OC and BD, OD are different, the angle is the same for 2 circles of differing sizes.
• We look at ratios and find that $\frac{AC}{OC} = \frac{BD}{OD}$
• AC and BD are lengths of arcs on the circle, and OC and OD are the radius of the circle.

Radians

• 1 full rotation angle = arc length / radius
• For 1 full rotation, arc length = circle's circumference, i.e. $2\pi r$
• $\frac{2 \pi r}{r} = 2 \pi$ radians.
• The unit of measuring angles, called radians, is intrinsically tied to the properties of a circle.
• An angle of 1 rad happens at point A on the circle when the arc length equals the radius of the circle.
• 1 full rotation = 360°
• 1 full rotation is also = $2\pi$ rad
• $2\pi \text{ rad } = 360^\circ$

Conversions

• There are about 6.28 radians in a full rotation. 360° = $2\pi$ rad
  $\text{1 rad } = \frac{360}{2 \pi} = \frac{180}{\pi} \approx 57.3^\circ$
• Degree $\rightarrow$ Radian: $\text{angle in radians } = \text{angle in degrees } \times \frac{\pi}{180}$
• Radian $\rightarrow$ Degree: $\text{angle in degrees } = \text{angle in radians } \times \frac{180}{\pi}$

Examples

• Convert to radians: 30°, 45°, 150°, 270°
• $30^\circ = 30 \times \frac{\pi}{180} = \frac{\pi}{6} \text{ rad}$
• $45^\circ = 45 \times \frac{\pi}{180} = \frac{\pi}{4} \text{ rad}$
• $150^\circ = 150 \times \frac{\pi}{180} = \frac{5 \pi}{6} \text{ rad}$
• $270^\circ = 270 \times \frac{\pi}{180} = \frac{3 \pi}{2} \text{ rad}$
• Convert to degrees: $\frac{\pi}{4}, \frac{\pi}{3}, \frac{10\pi}{3}$
• $\frac{\pi}{4} = \frac{\pi}{4} \times \frac{180}{\pi} = 45^\circ$
• $\frac{\pi}{3} = \frac{\pi}{3} \times \frac{180}{\pi} = 60^\circ$
• $\frac{10 \pi}{3} = \frac{10 \pi}{3} \times \frac{180}{\pi} = 600^\circ$

Arc Length

• Based on the angle definition, we derive a relation between radius and arc length:
• $S = r\theta$
• $\theta$ in radians (not degrees).

Examples

• Find the length of an arc of a circle with radius 10m that subtends a central angle of 30°
  $\theta = 30 = 30 \times \frac{\pi}{180} = 0.52 \text{ rad}$
  $\text{S} = r\theta$
  $\text{S} = 10 \times \frac{\pi}{6} \approx 5.2$
• A central angle θ in a circle of radius 4m is subtended by an arc of length 6m. Find θ in radians
  $\frac{6 \text{ m}}{4 \text{ m}} = 1.5 \text{ radians}$

Application of Radians: Earth's Radius

• Greek mathematician Eratosthenes (276-195 B.C) measured Earth's radius.
• Measured the distance directly down a steep well in Syene.
• 500 miles north, in Alexandria the rays of the sun created an angle of 7.2° to the zenith.
• $S = r\theta$
• $7.2 = 7.2 \times \frac{\pi}{180} \text{ radians}$
• $r = \frac{S}{\theta}$
• $r = \frac{500}{(7.2/180)*\pi}$
• $r = 39809$

Non-Uniqueness of Angle Measure

• An angle can be reached by larger rotations.
• Is due to the fact that after one rotation, the angle repeats.
• Example: 30° and 390° represent the same angle.
• Principal values are between 0° and 360° (or 0 and $2\pi$ radians).
• If not, reduce by subtracting multiples of 360°:
  • Example: 470° - 360° = 110°
  • Example: -50°, -50° + 360° = 310°

Calculator Usage and Negative Values

• Lengths can't be negative.
• Lengths can't be hypotenuse is always positive
• Calculating Trigonometric functions: Sin (71°), Cos (112°), Sin (157°), tan(217°), tan (711°)
• Results:
  • 0.945
  • -0.375
  • 0.391
  • 1.279
• What's up with the negative values?
• Values are based on quadrants.

Quadrants

• Q1 (0° - 90°): All trigonometric functions are positive.
• Q2 (90° - 180°): Only sine is positive.
• Q3 (180° - 270°): Only tangent is positive.
• Q4 (270° - 360°): Only cosine is positive.
• ASTC Rule:
  • All Students Take Calculus
  • Indicates the positive trigonometric functions in each quadrant.

Trigonometric Ratios and Real Numbers

• Circle of radius r = 1 (Unit Circle)
• $\sin \theta = y$; $\cos \theta = x$ for any point in the unit circle.
• What's the equation of a circle?
• $x^2 + y^2 = 1$
• $\cos^2 \theta + \sin^2 \theta = 1$

Standard Trigonometric Identities

• Reciprocal Identities:
  • $\csc \theta = \frac{1}{\sin \theta}$
  • $\sec \theta = \frac{1}{\cos \theta}$
  • $\cot \theta = \frac{1}{\tan \theta}$
• Quotient Identities:
  • $\tan \theta = \frac{\sin \theta}{\cos \theta}$
  • $\cot \theta = \frac{\cos \theta}{\sin \theta}$
• Pythagorean (or Circular) Identities:
  • $\sin^2 \theta + \cos^2 \theta = 1$
  • $1 + \tan^2 \theta = \sec^2 \theta$
  • $\cot^2 \theta + 1 = \csc^2 \theta$

More Trigonometric Identities

• Derive new trigonometric identities.
  Divide all expressions by $\sin \theta$ to get, the following expression.
  $\frac{\sin^2 \theta}{\sin \theta} + \frac{\cos^2 \theta}{\sin \theta} = \frac{1}{\sin \theta}$
  $\implies 1 + (\frac{\cos \theta}{\sin \theta})^2 = \csc^2 \theta$
• Divide all expressions by $\cos \theta$ to get, the following expression.
  $\frac{\sin^2 \theta}{\cos \theta} + \frac{\cos^2 \theta}{\cos \theta} = \frac{1}{\cos \theta}$
  $\implies (\frac{\sin \theta}{\cos \theta})^2 + 1 = \sec^2 \theta$

Examples Cont.

• An expression for $\sin \theta$ terms of $\cos \theta$ where the angle is in each quadrant.
• Use $\sin^2 0 + \cos^2 0 = 1$
  $\sin \theta = \pm \sqrt{1 - \cos^2 \theta}$
• In Q1 Q2, the value is $\sqrt{1 - \cos^2 \theta}$
• In Q3 Q4, the value is $- \sqrt{1 - \cos^2 \theta}$
• Example: Express tan θ in terms of sin θ, where θ is in Quadrant II.
• express $\tan \theta = \frac{\sin \theta}{\cos \theta}$
• But $\cos \theta = \pm \sqrt{1 - \sin^2 \theta}$
• Therefore in quadrant II then $\cos \theta = - \sqrt{1 - \sin^2 \theta}$
• Therefore, $\tan \theta = \frac{\sin \theta}{-\sqrt{1 - \sin^2 \theta}}$

More examples

• Example: If tan θ = 2/3 and θ is in Q3, find $\cos \theta$
• So cos θ = 1/$\sqrt{ 1 + (\frac{2}{3})^2}$
• = 1/$\sqrt{ 1 + \frac{4}{9}}$
• = 1/$\sqrt{ \frac{13}{9}}$
• Since it is quadrant 3, $\cos \theta = - \frac{3}{\sqrt{13}}$
• Since it is quadrant 3, $\cot \theta = - \frac{3}{\sqrt{13}}$
• Example: If sec θ = 2 and θ is in Q4, find other 5 trig functions of θ.
• $\cos \theta = \frac{1}{2}$
• $\sin \theta = -\sqrt{(1 - \frac{1}{4})} = \frac{-\sqrt{3}}{2}$
• $\tan \theta = (\frac{-\sqrt{3}}{2})/ (\frac{1}{2}) = -\sqrt{3}$
• $\csc \theta = \frac{-2}{\sqrt{3}}$
• $\cot \theta = - \frac{1}{\sqrt{3}}$

Trigonometric Functions

• Can take the leap from trigonometric ratios to trig functions.
• $y = sin(x)$
• Takes 'angle' as input
• Returns 'real number' as output
• Example: $y = sin (x)$
• x = 32°
• y = sin (32) = 0.53
• $y = cos(x)$
• x = 32°, y = 0.85
• $y = tan(x)$
• x = 32°, y = 0.62

Domain and Range for Trig Functions

• Focus on Sin θ and Cos θ
• Use right triangle and also unit circle analogies.

Range for Right Triangle

• What is the least value $\sin \theta$ can take?
• As y approach 0, $\sin \theta$ approach 0.
• What does the triangle look like, it becomes a line
• What is the most value $\sin \theta$ can take?
• Now, the y = r, or also know as the, radius; then $\sin \theta= 1$
• What happens to the angles its 1.

Circle Explanation?

*What is the least value $\cos \theta$ can take?
*What does the triangle look like, it becomes a line
*What is the most value $\cos \theta$ can take and what happens to the angles when its 1.
*

• Now consider the same things, but on a circle rather than a right triangle:

Cos and Sins impact on graph?

*

• What happens to $\sin \theta$ (or $\cos \theta$) as the point P moves counter-clockwise around the circle.
  Analysis:

Now consider the
same things, but circle
rather than a

• Quadrants
  radius stay
  the same values
  of sin O repeat 4
• Times
  point (x,y)
  will
  have
  corresponding
  point on
  each quadrant
  resulting in a square shape and also that the radius
  stays the same resulting in a square shape. Therefore the values of $\sin \theta$ repeat four times.

Function Graphs and Properties

• Found that the values for $\sin \theta$ repeat in an interesting pattern with the four quadrants. That’s because the function is called periodic because the behavior repeats
• As y is capped at 1 and bottomed at -1
• Now since domain is all real numbers we can also say its ( - infitiy, infinity)
• Found is the also periodic its period is $2\pi$ or 360.
• The properties from the analysis, therefore the function is called periodic because the behavior repeats. what exactly would we call this?

Table of Values & Summary

Now we know all this draw conclusions on the periodic table

• Now that we knew the fundamental properties of these trigonometric functions, we can make a table and which are actually plot the functions.
• Now, we Fill in
  the table, that describes the periodic functions.
  x =
  0, \frac{π}{6}, \frac{π}{3}, \frac{π}{2}, \frac{2π}{3}, \frac{5π}{6}, π ; \frac{7π}{6}, \frac{4π}{3}, \frac{3π}{2}, \frac{11π}{6}
  π ; \frac{7π}{6}, \frac{4π}{3}, \frac{3π}{2}, \frac{11π}{6}

Values and Notes

• y = $\sin \theta$ and values for all those points
• 0; \frac{1}{2}; \frac{{\sqrt{3}}{2}}; 1; \frac{{\sqrt{3}}{2}}, \frac{1}{2} , 0 ; -\frac{1}{2} -
  \frac{{\sqrt{3}}{2}}; -1 ;\frac{{\sqrt{3}}{2}}, -\frac{1}{2}
  Cos \theta and values for all those points
  1; \frac{-\sqrt{3}}{2 \frac12; -1}; 0; \frac{\sqrt{2}}{2}} : -1\frac{\sqrt{3}}{2}}; 1
  Also note, to find the y and \cos\theta values you have to memorize these values.
• The graph repeats after 2+\pi$$
• This can be visualized
  corresponding graphs.