PreCalculus Notes

Ch.5.1: Angles & their measure

• When 2 rays are drawn w/ a common vertex, they form an angle. One ray of an angle is called the initial side & the other is a terminal side
• An angle θ is said to be in standard position if its initial side is on the positive x-axis and its vertex is at the origin.

Quadrantal Angles

Angles whose terminal side lies on the x-axis or y-axis are called quadrantal angles. Examples include 0°, 90°, 180°, 270°, and 360°.

• Initial side is on the positive x-axis.
• Terminal side:
  • Above = Positive
  • Below = Negative

Angle Conversions

• Full revolution: $360°$ or 1 full revolution or $2π$ radians.
  • $135° = 90° + 45°$
  • $495°$ is an example of an angle greater than 360°.

Minute, Second Conversions:

• Degrees, Minutes, and Seconds:

  • Conversions:
    • 1 minute = 60 seconds
    • 1 degree = 60 minutes = 3600 seconds

• $65.917°$ to degrees, minutes, seconds:

  • $0.917 * 60 = 55.02$ minutes
  • $0.02 * 60 = 1.2$ seconds
  • So, $65.917° = 65° 55' 01.2"$

• $65° 5' 17"$ to decimal degrees:

  • 5/60 = 0.0833 degrees
  • 17/3600 = 0.00472 degrees
  • So, $65° 5' 17" = 65.0879°$

Central Angles

• A central angle is a positive angle whose vertex is @ the center of the circle.

Arclength Formula

• Arc length: $s = rθ$
  • s = arc length (any unit)
  • r = radius (any unit)
  • θ = central angle (radians)
• $s = rθ = s/θ → θ = s/r$
• Example:
  • $s = 4m, θ = 0.75$
  • $s = 3m$

Radian Conversions

• $1 degree = π/180 radians$
• $1 radian = 180/π degrees$
• $1 revolution = 2π radians$
• $180° = π radians$
• '*' will always mean radians
• $180°$ always mean degrees.
• $\frac{π}{6} rad = 180°$
• $\frac{π}{6} rad =30°$
• Conversions Summary degrees to radians!

Unit Circle

• Diagram of the unit circle with degree and radian measures, and corresponding coordinates.

Area of a Sector

• $A = \frac{1}{2} r^2 θ$
  • r = radius
  • θ = angle (in radians)
  • Example: Find the area of a sector with radius $r = 6$ and angle $\theta = \frac{4π}{9}$

Linear and Angular Speed

• Linear Speed: $v = \frac{s}{t}$

  • s = distance (arc length)
  • t = time

• Angular Speed: $ω = \frac{θ}{t} = \frac{velocity}{linear speed}$

  • θ = angle (in radians)
  • t = time

• Formula relating linear and angular speed: $v = rω$

Trigonometric Functions: Right Triangle Approach

• Right triangle definitions:
  • $\sin B = \frac{opp}{hyp}$
  • $\cos B = \frac{adj}{hyp}$
  • $\tan B = \frac{opp}{adj}$
  • $\csc B = \frac{hyp}{opp}$
  • $\sec B = \frac{hyp}{adj}$
  • $\cot B = \frac{adj}{opp}$
    *Mnemonic : SOH CAH TOA
• Also applies:
  • $\csc = \frac{1}{\sin}$
  • $\sec = \frac{1}{\cos}$
  • $\cot = \frac{1}{\tan}$
• Applying to Angle A:
  • $\sin A = \frac{opp}{hyp}$
  • $\cos A = \frac{adj}{hyp}$
  • $\tan A = \frac{opp}{adj}$
  • $\csc A = \frac{hyp}{opp}$
  • $\sec A = \frac{hyp}{adj}$
  • $\cot A = \frac{adj}{opp}$

Unit Circle

• Coordinates of points on the unit circle for reference.
• Sine Function:
• Sine Function associates w/ the y-coordinate of P & is denoted by$\sin θ = y$
• Cosine is on the x-coordinate of P & is denoted by $\cos θ = x$
• tangent associates wit; the ratio of the y to the x-coundinate of P & is denoted by $\tan θ = \frac{y}{x}$
  • There are diff unit circles for sin, cos, tan, csc, sec, cut. Examples are below:

Pythagorean Theorem

• $x^2 + y^2 = r^2$

• $\sin θ = \frac{y}{r}$, $\csc θ = \frac{r}{y}$, y ≠ 0

• $\cos θ = \frac{x}{r}$, $\sec θ = \frac{r}{x}$, x ≠ 0

• $\tan θ = \frac{y}{x}$, x ≠ 0, $\cot θ = \frac{x}{y}$, y ≠ 0

• Right Triangle:

  • Opposite side squared plus adjacent side squared equals hypotenuse squared.

Example

• Hypotenuse is 24, adjacent side is 12. Find the opposite side:
  • $24^2 = 12^2 + b^2$
  • $576 = 144 + b^2$
  • $432 = b^2$
  • $b = \sqrt{432} = 12\sqrt{3}$

Coordinates Example

Coordinates: (4, -3)

• sine = $\frac{y}{r}$
• -3/5
• Cosine = $\frac{x}{r}$
• 4/5
• (4,-3) $\sqrt{4^2 + (-3)^2} = \sqrt{25} = 5$
  *If we're at the 4th quadrant, then we adjust b/c y is negative.
  Always Remember SOH CAH TOA

Even-Odd Properties

• Sine is odd:
  • $\sin(-θ) = -\sin θ$
  • $\csc(-θ) = -\csc θ$
• Cosine is even:
  • $\cos(-θ) = \cos θ$
  • $\sec(-θ) = \sec θ$
• Tangent is odd:
  • $\tan(-θ) = -\tan θ$
  • $\cot(-θ) = -\cot θ$

HW Examples

Section : Trigonometric Functions of Acute Angles

• Key Concepts
  • Trigonometric functions relate angles to side lengths in right triangles.
  • The six trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent.

Exact Values for 45 Degrees ($\frac{π}{4}$)

• For a 45° angle, the triangle is isosceles, leading to equal side lengths.

• Using the Pythagorean theorem, the lengths of the sides can be determined as I and 2 for a triangle with hypotenuse 1.

• The exact values are: $\sin (45°) = \cos(45°) = \frac{\sqrt{2}}{2}, \tan(45°) = 1$

• 45°

  • c = $\sqrt{2}$
  • b = 1
  • a = 1

Exact Values for 30 and 60 Degrees ($\frac{π}{6}$ and $\frac{π}{3}$)

• For 30° and 60°, a right triangle can be constructed with a hypotenuse of length 2.

• The side opposite 30° is 1, and the side opposite 60° is $\sqrt{3}$, derived from the properties of 30-60-90 triangles.

• Exact values are: $\sin(30°) = \frac{1}{2}, \cos(30°) = \frac{\sqrt{3}}{2}, \tan(30°) = \frac{\sqrt{3}}{3}$

• 30°

  • c = 2
  • b = $\sqrt{3}$
  • a = 1

• 60°

  • c = 2
  • b = $\sqrt{3}$
  • a = 1

Section 2: Using Calculators for Trigonometric Functions

• Calculators can approximate trigonometric function values for any angle.
  *Ensure the calculator is set to the correct mode (degrees or radians) before calculations.

Section 3: Applied Problems Involving Right Triangles

• Real-world problems can often be modeled using right triangles and trigonometric functions.
• Common applications include construction, surveying, and meteorology.
  Ex. Constructing a Rain Gutter
  Steps for Using a Calculator
  Example: Finding the Width of a River

Section 4: Angles of Elevation and Depression

• The angle of elevation is the angle from the horizontal up to an object.
• The angle of depression is the angle from the horizontal down to an object.
  Ex. Height of a Cloud
  Ex. Height of Statue

5.3 Homework

• Sine (45) = $\frac{-\sqrt{2}}{2}$, so $\csc(45) = -\sqrt{2}$

5.4 Lecture

Pythagorean Identities
Sine Function y = \sin x, -∞ < x < ∞

• * Odd Function
• Max is 1 & Min is -1

Cosine Function

• Same period & range as sine
• If you shift the cosine If to the right, it turns into a sine function.
  *Horizontal Transformations

Sine and Cosine Functions Amplitude & Period
Amplitude = |A|
Period = $2π/w$
(where w is the angular frequency)
2 2
y=sin(x) * Stretches; Doubles the period (4π)
Sine Reflected
y=-3 sin (w= \frac{2}{5}x)
The Unit Circle
Degree Radian (x,y) Function

The Unit Circle Chart:

• C-S Romito O
• sin from Otol
  *Reference Angle
  *Coterminal Angle: Diff angles that have side.
  Complimentary sides

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  *Graphing Transformations
• A central angle is a positive angle whose vertex is @ the center of the circle
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