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

Review Seer Chapter 5 &

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